Some of the main arguments I've heard and read are below. Have a scroll through them, or jump straight to one that interests you from these links:

Europe and the EU - The Scotch Whisky Industry - The British Pound - Shopping Prices - Alex Salmond - Finances - The UK Government - The Border - Nuclear - Energy and Resources
 

 Europe and the EU

The other part of this is that the No camp are claiming that it will be difficult for Scotland to rejoin the EU as a separate, independent country, primarily because other European countries would be afraid it would set a precedent for separatists in countries like Spain and Belgium making a bid for their own independence. What they fail to explain is that Scotland is not like these autonomous regions, Scotland is a country (yes, even now) and so already has a far more independent status than these regions which will obviously work in Scotland's favour during any membership application to the EU.

This, to me is FUD, selective dissemination of information by the No camp in order to promote fear, uncertainty and doubt. In reality, this end of the argument is all speculation as nobody in the Yes or the No camps really know what will happen. I personally don't think there will be all that much of a problem joining the EU once Scotland take on the Euro. And to be honest, that wouldn't be the big disaster that some people seem to think.
 

  The Scotch Whisky Industry

The no side is somewhat more puzzling to me. Yes, it's a £4.3 billion industry and that's very important to Scotland. That much I get, but here's where it falls down for me. The crux of the No argument is that Scotland needs the UK government in order to maintain relations with the export markets and to negotiate trade deals using their substantial clout. While this has no doubt helped in the past. Scotch whisky is a world renowned product, exported to as many as 200 countries around the world. If there is such demand for Scotch whisky worldwide, would people continue to want it from an independent Scotland? Or would all these countries not know where to import it from, now that the UK government doesn't represent Scottish produce any more? Something tells me the former is more likely here.
 

  The British Pound

Sterling appears as a funny curency when you look at it from a foreign perspective. Sterling notes aren't technically currency - they're basically an IOU from the bank that printed them (read the small print on any bank note if you don't believe me). And that's why, even now, you can't use Scottish bank notes in shops in England. Not only that, but technically speaking, Bank of England banknotes aren't legal tender in Scotland either. So it's not as big a deal to imagine what it might be like with a separate Scottish pound. Scotland already has a mostly separate currency of its own, the Scottish Pound.

As an aside, Ireland started off with its own pounds after its split from the UK. Yes, they weren't as stable as the Sterling pound, but the Irish economy was much smaller than the Scottish economy, and once it stabilised, happily coexisted in the world with Sterling. The numbers didn't match (it cost around 7 Irish pounds for 5 Sterling pounds), but the prices in shops were out by a similar magnitude, so the actual value was much closer than the bare numbers would lead you to believe.

Another possible outcome of not having a currency union (and most likely a requisite for rejoining the EU) is that Scotland adopts the Euro as its currency. The No camp seem to think this would be a really bad thing, and that Scotland should vote No in order to avoid taking it on, opting to stay with the pound and as a result, a part of the sixth largest economy in the world. But if you look at the Euro at the moment, it's really not doing as badly as the No camp information would lead you to believe. The might of German industry and the not-to-be-underestimated French economy are both larger economies than the UK, and both are based on the Euro. That gives the Euro a lot of stability and power - in fact, the Euro to Sterling exchange rate is almost exactly the same as it was when the Euro was introduced over ten years ago. And that's even after multiple bailouts issued by the ECB to some member countries with economic difficulties. Stability shouldn't be any sort of a concern regarding the Euro.

On a personal note, coming from a country that switched to the Euro, I personally found no problems with the switch. My savings didn't suddenly become worthless, I could still afford to buy the very same shopping I did when we used the pound, and my wallet was in better condition as the coins are physically smaller and lighter, and so there was less bulging under the strain of a night's worth of pub shrapnel!

The Yes camp on the other hand, is convinced that a currency unit can be agreed upon because it would be in the interest of the rest of the UK to keep it that way. After all, with such huge revenue generated by Scotland's many important industries (such as oil, gas, manufacturing, whisky...), it would make sense to keep these markets stable. That way, the UK can continue to enjoy these Scottish exports without having to worry about exchange rates potentially increasing the cost of fundamental products like energy and food. Of course it could go both ways - Scottish exports could drop in price too - but with Scotland's economy growing and Scotland exporting almost half of its entire oil and gas production to the rest of the UK, it's unlikely that prices would fall by much, if at all.

And, as I've said above, even if it doesn't happen and Scotland goes for the Euro instead, it wouldn't even be close to the end of the world.
 

 Shopping Prices

Besides, ask anyone in the Republic of Ireland to choose between paying a little bit more for their groceries and being a part of the UK, and they'll go for the more expensive groceries every single time. Price differences like this are a minor grumble in countries like Ireland - I don't see why Scotland would be any different, especially within a strong economy.

Update: Tesco have responded to the claim of price rises printed in the No campaign leaflets. They state that this information is purely speculative, and that they are maintaining a neutral position on Scottish independence. They did add however, that certain products are actually cheaper in Ireland than in the UK.
 

  Finances

Another place where figures don't match between both sides is regarding pensions. The No side are claiming that pensions would be in danger (more apparent threats) if Scotland voted for Independence, but the Yes side are saying that they would be perfectly safe because an Independent Scotland would have more control over them and more security coming from the fact that it has a higher GDP and hence a stronger economy than the rest of the UK, so it would easily be able to afford a protection fund similar to that currently run by the UK government.

The thing with these figures though, is that neither side are particularly convincing. Both could be making them up, and I've no reason to believe either side over the other. But if both sides' claims are taken at face value, it doesn't look like things are as doom and gloom as the No side would have you believe.
 

 The UK Government

Speaking of democracy, I know many British people are fond of the Royal Family, and that's perfectly fine. They're not elected, but they don't have much sway in the day-to-day running of the country either so they're effectively a strange sort of tourist attraction. What does seem utterly bizarre to me however, is that the House of Lords exists. Coming from the Republic of Ireland, the very idea of one of the main houses of parliament consisting of unelected, inherited and male-dominated seats is completely, well, foreign. They are people that actually do have an influence on every day life in the UK, yet the ordinary people don't have any say whatsoever in who they are. Hell, there are 26 bishops in the House of Lords, there by no other virtue than the fact they are bishops. I'm sorry, but that's an embarrassingly antiquated way of operating a so-called democracy. An independent Scottish government would be formed entirely from elected members - no inheritance of titles or seats, no nepotism. Or at least that's the theory. There will always be corrupt politicians whether Scotland stays with the Union or becomes independent. But at least an independent Scotland will have a chance to vote them out.
 

 The Border

Here's another one I've seen bandied about a couple of times. The No campaign have given an example of an English family who have some members living in Scotland. One quote from them reads: "We all find it ludicrous that they would need to use their passport to visit friends and family at their home from home." Now this has me stumped. Will Scotland have to erect a closed border to the rest of the world if they became independent? Or will the rest of the UK have to border up Scotland to keep migrant Scots from coming in? Well, if they insist on being a separate country...

But here's the thing: I didn't need a passport to visit my friends and family in the UK from Ireland when I was living there. And Ireland hasn't been part of the UK for over 90 years. So why would Scotland have to have a closed border with passport controls when Ireland doesn't? Perhaps it's because, by leaving the UK, Scotland would be leaving the EU as well? Maybe... But then again, look at the borders between Switzerland and the EU countries it neighbours: they're all open. And the same goes for Norway. Neither of them are in the EU, yet there is free movement across their borders with their EU neighbours. Truth is that while there might be an actual border erected, it will be no different than that between any other neighbouring western European countries.
 

 Nuclear Power and Weapons

Next up is nuclear power. People are cautiously wary of this. It provides huge amounts of energy for very little (albeit extremely hazardous) waste, does not produce greenhouse gases, and while not renewable, is almost limitless. There are generally legitimate safety concerns, especially with the Fukushima Daiichi incident still fresh in people's memories, but on the whole it's clean, safe and reliable when treated with the respect it demands. Nevertheless, renewable, hazard-free energy is always a better option, and that's an area where Scotland are continuously and rapidly expanding. More about power and energy later, but the basic points are that Scotland produces around 15%-25% more power that it uses. Where does all that electricity go? Onto the UK grid for consumption across the rest of the UK. Roughly one third of Scotland's electricity is produced by nuclear power stations, so that's about 10% each. An independent Scotland would have full control over power production in Scotland, which would allow it to decide what to do with that extra 20%. The way I see it, three things could happen: First, Scotland could continue to export to the UK, generating revenue for the country from abroad. Or second, Scotland could stop exporting power, which would allow it to shut down two of the three nuclear power plants currently operating in Scotland - and that's without building a single new wind turbine. And finally, Scotland could use that extra power in the place of power currently generated by fossil fuels which could then be exported to generate revenue for an "oil fund". Either way, it would be Scotland's decision and Scotland's decision alone, and it's actually a comfortable position to be in.

My last nuclear-related point, and one which I've only heard mentioned once in the campaigns for the referendum, is to do with Sellafield reprocessing plant and its related nuclear facilities. This plant was built primarily as a manufacturing facility for weapons-grade plutonium, not electricity, and as a result was never designed to be cost-efficient or socially responsible. Decades of poor management, cost cutting, buyouts and subsidising by the UK government have left the facility in a decrepit state. Shortcuts taken when it was first built, and during every overhaul, refit and patch job have left Sellafield in such a sorry state that when it finally became time to decommission the facility and clean up the mess, it was estimated that it would take decades of work and several billion pounds to finally see the area in a fit state. Since the work has started, the estimated decommissioning time has been increased to 120 years, and the cost of decommissioning to £70 billion! That's more than half a billion pounds of tax payers' money each and every year for 4 generations! One hell of a bill for some antique nuclear weapons and a small amount of electricity that was sold below cost from a plant that never made an operating profit in its lifetime. It could be argued that this is partially Scotland's responsibility as they were involved in World War Two as a part of the UK when the decisions were made to build this plant. It's so long ago though that it's possible nobody knows whether Scotland had any say in the matter at the time. Personally though, I don't think Scotland should pay for this mess any more, and Independence would allow Scotland to decide whether to keep paying for it, or to leave it for the government who created it in the first place to take care of.