When wines get to be over a decade old, you sometimes need to treat them differently to wines that are younger. Here are a few of the common issues people have, and how to deal with them.

Corks

Corks are bits of tree bark that seal the wine bottle. As a natural product, no two corks are alike, so here are some of the issues you may find with corks, most of which won't make the wine taste any different.

Corks are sticking - This is a good thing (sort of!). It means that the cork is still springy and therefore holding a seal.  Using a T-shape corkscrew or a butlers thief (see corkscrew guide below) will get the cork out. If you can't get it out, or if it rips apart, just shove it down into the bottle and decant the wine.

Corks are breaking - Usually, this will have no issue on the wine, it is just that the cork has broken. The wine is not 'corked' if the cork breaks (see flaws). Again, choosing the correct corkscrew will help this not happen. Please note that we will not exchange/refund bottles of wine where the only issue is that the cork is stuck or has broken, as we have no way of knowing how you have approached opening the bottle. Think of it this way, if you bought a car, went out on the road and damaged it by crashing into a lamp post, you wouldn't expect the car dealer to take the car back because you'd wrecked it. The same applies with wine. If the wine has gone bad (see flaws) then obviously, we will happily refund or exchange it for you.

Cork is mouldy - The cork being mouldy on the top isn't a sign of the wine going bad. It might be an issue, but the point of a cork is to act as a seal, so if the outer part of that seal is mouldy, logic dictates that it won't necessarily be on the inside where the wine is.

Cork has a line of wine down it - Now this could be an issue. This means that there is a weak spot in the cork and wine has seeped through the cork. The wine may be oxidised, but might not. Taste the wine and if it is nice, great, if it is vinegary then it is bust.

The cork is very very wet all the way through - Again, this could be an issue, with the wine being oxidised. A vinegary flavour will indicate if the wine is bad or not.

Flaws that are and flaws that aren't

Firstly, remember it is incredibly unlikely that any flaw with a wine will make you ill. It may taste awful, but unlike most consumables, wine is usually perfectly safe to drink once it has gone off. You'd just be a bit of a nutter if you chose to do so! Here are the main flaws that can happen to wine, and some that aren't flaws as well.

Smells like wet dogs or cardboard - This is when a wine is corked, where the wine has reacted with the cork and gone off. It is beyond salvation and bring it back for a refund or swap.

Smells vinegary - This is oxidisation, caused by the wine being exposed to oxygen for too long. If you leave a bottle open for too long, this will happen, but in a freshly opened bottle it shouldn't be an issue. If it is, then bring it back for an exchange or refund.

Smells eggy - This is sulphur compounds in the wine. You can sometimes get rid by decanting the wine (even white or sparkling) and stirring with silver. If that doesn't work, bring it back.

Bubbles in a non bubbly wine - This is secondary fermentation. They can smell yeasty. Some residual sugar from the juice has remained and fermented once the wine has been bottled. This should just be brought back.

Heat damage - If you keep your wine next to a fire, cooker, washing machine, in a boiler room, in an attic, in an overly hot house or (and I have heard this once) in a greenhouse, you are going to wreck your wine and cook it. This is usually coupled with a burnt sugar like aroma. Now it won't happen in a day or week, but if you buy the wine and then return it months later after keeping it in poor conditions, we're going to possibly question how it has been kept if this is the flaw.

Bits in the wine - This is either tartrate crystals (if it looks like glass shards), or a dark sediment (dark, purply grit). These are totally fine, you just need to filter them out. The tartrate crystals come from high minerally wines (usually white) and the sediment is all the extraction from the wine being in content with grape skins solidifying and falling to the bottom of the bottle. Both are harmless and not a flaw. A lot of winemakers (usually with long beards) are now producing wines that aren't filtered or fined (natural wines), so you are finding more and more wines with dead yeast (lees), bits of grape skin, twigs, branches, hair from their beard.... (the last few were a joke) - this is apparently how they want their wines to be consumed, so don't panic.

Corkscrews

What corkscrew you use is INCREDIBLY important for each bottle of wine. But regardless of what you use, if the cork breaks during the opening process, just shove the remains into the bottle (if you can't get them out) and then put the wine through a sieve to filter out any bits. The wine will not be ruined if the cork breaks.

Winged Corkscrew - The majority of winged corkscrews are about as much use as a chocolate teapot. Everyone has them and most of them are comprised of a solid metal shaft with a lip coiling around them that is supposed to grab onto the cork. In reality, it drills a hole through the cork and when any pressure is applied, it just rips a hole in the cork. The only exception to this is if the actual screw looks like a zig zag not a drill. The black version shown is what you want, the silver one is not.
T Shaped Corkscrew - This is the original, still the best, however you do need muscles to haul the cork out and it is sometimes a two person job if the wine is old. Even if the cork breaks, you can usually fish the remaining bits out of the bottle.
Waiters friend - Good for the majority of wines (as long as they have the zig zag screw part as shown) but with older wines the fact that they lever the cork with a lateral movement not just vertical can put undue stress on the cork. Often better to just use it as a T shape corkscrew once you have used the lever action getting the cork moving a little. Again, if it breaks, just use it to fish out the bit remaining.
Butlers Thief - The two prongs go down the side of the cork and hold it together. (see youtube video to show what to do). These are perfect for sticking corks or older wines. There is a fancier one called a Durand that combines a Butlers Thief and a T shaped corkscrew. I've never used one (yet) but they are apparently fantastic for older wines. This is my default corkscrew now.
Screwpull - These work very very well with most corks (except plastic), but occasionally they can put too much pressure on a stuck cork and you may think you're going to break the cork or the expensive corkscrew. The major advantage is that they have a vertical pull that is controlled. I do have one, and it is very good and very quick to use.
Flavour of older wines
Because wine is a living being, it changes over the years. As wine gets old, they change, evolve and taste different from how younger wines do. Because most people don't have wine cellars, and they tend to drink the bottle they purchase within a few days, most people are rarely exposed to older wines, so don't know how they taste.
Reds tend to get lighter, the fruit can become less prominent and more savoury flavours emerge. The wines will develop a style that is less 'sit back on the sofa and watch telly' and more a 'lets have a nice meal and pair this with a haunch of venison'. Therefore if you are having a meal with the wine, older wines can be exemplary, but if you want a 'sit back and enjoy a glass' wine, older reds might not be for you. 
Whites tend to fatten up with age, and again, that bright, fruit forward style will subside over time, and the wine becomes more food friendly. Whites like Kiwi Sauvignon Blancs can age brilliantly, but they will change massively and may not be what you are looking for when it comes to your choice. Aged Burgundy is phenomenal but is more than likely a bad choice for anyone just wanting an easy glass of wine due to the complexity of it. 
Champagne, port, sweet wines - they all evolve too and can age for decades, even centuries, so it is probably better to ask us about older wines before you buy them.