I put stuck in quotes above, because #1 and #6 are not really stuck ferments. They are frankly, what is expected. The order of the reasons, is based upon the frequency I've encountered among my fellow homebrewers.
First of all, if the yeast poops out you should first determine if you are close to the expected apparent attenuation for the yeast (see the yeast FAQ). If you are, then nothing you will do, short of adding a more attenuative yeast will get the gravity to drop any further. You calculate the apparent attenuation by using the following formula:
%ApparentAttenuation = (1 - (FG - 1)/(OG - 1)) * 100
So, for example, if your OG was 1.050 and your FG was 1.012, then:
%ApparentAttenuation = (1 - (1.012 - 1)/(1.050 - 1)) * 100 = (1 - 0.012/0.050) * 100 = (1 - 0.24) * 100 = (0.76) * 100 = 76% apparent attenuation
The most common cause is really a very unfermentable wort. Usually this is caused by using a malt extract such as Laaglander dry or "Dutch" dry malt extract. With these two extracts, I have known brewers to experience apparent attenuations of only 55%.
Insufficient nutrition is common only among beers that are made of very large percentages of corn sugar, cane sugar, rice sugar or honey. When making meads, some kind of yeast nutrient is virtually mandatory. The best solution, in my opinion, is to decrease the percentage of these nutrientless sugars and use more malt, but the easiest solution is to add yeast nutrient. Chances are that, even if you are trying to make an American Light Lager, your beer will taste a lot better if you don't overdo the adjuncts.
Temperature shocks (sudden changes, downward usually, in temperature) can cause yeast to flocculate out early and settle on the bottom. For example, Wyeast American Ale (#1056) really doesn't like temperatures much below 63F. At 57F Wyeast #1056 will simply sink to the bottom and go to sleep. The solution is to warm up the batch into a comfortable range for the yeast you used (ale or lager) and then stir the yeast up into suspension again somehow (without aerating). A better solution would be to warm up the batch and pitch a new starter. A batch I recently made with Wyeast #1056 which I accidentally let get down to 57F, never quite got restarted after I warmed and swirled, finished a little high and then overcarbonated in the bottles.
If you are trying to make a barleywine or a triplebock and your yeast poops out, you must ask yourself: "Did I aerate enough?" and "Did I use a big enough starter?" There's no way for me to tell you how much aeration is enough, but I use at least a 2 liter starter for OGs over 1080 or so and a 4 liter starter for OGs over 1100 or so. I don't actually pour all the spent wort from the starter into the batch, but rather, I make a 2 liter starter, let it ferment out, pour off the spent wort, add 2 liters more wort let that ferment out, pour off most of the spent wort and use that. If you answered no to either of those two questions, you can make up a new, bigger starter and pitch that.
If the alcohol level is really high, then about all you can do is to pitch a more alcohol-tolerant yeast, such as Pasteur Champagne yeast, available both dry and from Wyeast. You should still make up a big starter even if you use a more alcohol-tolerant yeast. Sometimes rousing (stirring up the yeast) can lower the gravity a few more points (try to not aerate the wort too much or you will elevate diacetyl levels).
If you are using a very highly flocculent yeast, such as that used by Samuel Smith's brewery, then you will have to keep rousing the yeast to get it back into suspension.Al Korzonas BJCP Master Judge Owner, Sheaf & Vine Brewing Supply
Spencer W. Thomas (email@example.com)