Yeast
There are a number of yeasts you can use to make mead. I've used various champagne yeasts a lot, as I don't make my meads incredibly strong, and I like a dry, sparkling mead. The drawback of champagne yeast is that it's not too tolerant to very high original gravities, and it is very tolerant to high alcohol content. This means that it will start very slow if you're brewing a high-gravity mead, but it will ferment out almost all the sugar available. If you're brewing high-gravity meads, I'd suggest starting with either a wine or mead yeast, and if you like a dry, strong mead, finishing it with champagne yeast. If you like a sweeter mead, you can use a mead or a wine yeast right from the beginning.
A major development in the three decades since I initially wrote Mead Made Easy is knowledge about kveik. Kveik, briefly, is an heirloom yeast, typically used in farmhouse brewing in western Norway. It typically works at much warmer temperatures than traditional brewing yeasts, and also works faster. A friend made a traditional mead which was ready in a month. Kveik yeasts also typically can handle higher alcohol levels than traditional brewing yeasts, as well. The main thing is that kveik needs a good quantity of nutrients. If you'd like to know more, Lars Marius Garshol's book, Historical Brewing Techniques: The Lost Art of Farmhouse Brewing is an excellent resource, as is his blog.
Some yeasts that I've had good results with, or have heard recommended by others, are Lalvin K1V, which is a fast starting, intermediate finishing wine yeast; Flor Sherry yeast, for a sweet finish (or you could use more honey with the Lalvin K1V); and Prise de Mousse, which is similar to champagne yeasts in finish. I've heard of people having good results with Vierka mead yeast, but I've had mixed results. I've also tried YeastLab Dry Mead yeast, which worked nicely. I think the problems I've seen with the Vierka may have been due to older yeast. As mead continues to increase in popularity with homebrewers, more new varieties should become available, and they'll be fresh more often.
Be aware that this list is far from complete, and there are probably many other suitable yeasts out there. For lower gravity meads, I've used Edme ale yeast with good results, and it's becoming a favorite of mine. As long as you don't make the mead with such a high starting gravity that the yeast never get a good start, results are pretty good, and slightly higher-than-normal temperatures don't bother ale yeasts as much as they do other strains. An added benefit is that the mead doesn't need as much aging as mead made with wine or champagne yeasts do, and you can enjoy it earlier.
In short, don't be afraid to try something new, but make sure you check the expiration dates. Some mead yeasts sell slow enough that keeping fresh stock in a homebrew store is a challenge. You may also want to check out the Zymurgy special issue on yeast and beer, or the Summer 1994 issue, which has a table listing yeasts and their characteristics. They describe a large number of yeasts, many of which will work well for mead-making.
Lallemand also has good information available on their web pages. In particular, when asked about using their yeasts in meads, they suggested that the K1V-1116 produces a more “light and fruity” product. ICV-D-47 produces fewer esters and more earthy tones, which are good for dry mead and cider that is being considered for aging on lees and perhaps in oak. It also enhances mouth feel by the production of trace amounts of 90,000 mw polysaccharides. The K1V would be great for light and fruity mead or cider with a little residual sugar.
The 71B-1122 would be an excellent choice for the American style, light and fruity mead and cider. It will produce a little more alcohol in cider, because it converts some of the malic acid in the apple juice to alcohol (different than bacterial malo-lactic fermentation).
WYeast 3184 (sweet mead) has gotten a lot of criticism on the The Mead Lover's Digest over the years. Some of this is justified, since it's a slow fermenter in many cases, but it can be made to produce a good mead. The most important thing to remember with 3184 is that you need a good amount of nutrients for it, and it does poorly in high-sugar or low pH conditions. The best results have come when people start with a small amount of honey, and feed additional honey into the must as fermentation slows. That is, you start out with honey and water for a fairly weak mead (say two to three pounds per gallon), and when the fermentation slows, check the pH, correct if necessary (3184 is good down to about 3.6), and then add the equivalent of a pound or less of honey per gallon of must. I discuss this in more detail in the section on High Gravity Mead.
Some have attempted to brew using wild yeast. Personally I haven't tried that, and my understanding is that your results will be highly dependent on your location. But if you are interested in harvesting wild yeast the article I just linked may be helpful.
Another thing to consider if you're having problems with slow fermentation is that you need to have a good strong starter solution with many yeasts. D-47 and 3184 are both examples that will not do well if you have a low yeast-cell concentration in the must. Also, if the sugar concentration is very high, almost no yeast will multiply well. Again, quoting Clayton Cone at Lallemand:
EC-1118, K1V-1116 and ICV-71B are recommended for ice wine and late harvest wine. The key to fermenting high brix must—30-40 brix—is cell population and lots of nutrients. No wine yeast grows well in a high sugar environment so you must add at least 1.2 g. of yeast/l of must and 0.5+ g. nutrients/l. Agitate the fermenting must frequently. The fermentation can be completed in <30 days.