There is a lot to hear about using horn shavings as ecological alternative to conventional fertilizer.
I was wondering - since my hair and nails are basically the same substance, could I use them in the garden?
Are there any possible problems?
Best not used directly as fertilizer around plants, but its fine to add both as a component of your compost heap. Both take a while to break down, and large quantities might cause problems, because human hair and nails can contain heavy metals from the environment which have entered the body. Whether toxic levels of heavy metals are present in hair and nails can only be determined by a laboratory, but if they were that high, the person would likely have some health problems, certainly over time. Some information in regard to hair clippings and their usefulness as a direct fertilizer here https://www.livescience.com/5260-human-hair-good-fertilizer.html
Hair does not breakdown for years and would make a fibrous mess in mulch. It does have a good garden use,however : It is very good to discourage rabbits from eating your garden. Apparently after months of exposure , rabbits still recognize it as evidence of a dangerous predator ( a k a , people).I used it sparingly in the fall ( to protect stems above the snow) and by spring most had blown away so little clean-up was needed.
Both hair and feathers are considered to be a slow-release source of nitrogen (I've read about this in the past, although the links in my answer here aren't my initial sources). They aren't significant sources of phosphorus or potassium, however.
Bamboo already alluded to this, but one study seems to suggest that human hair makes a reasonable fertilizer for some kinds of plants. I imagine fingernails have similar properties as hair, although they probably break down faster. I could be wrong. I looked and didn't find any sources on fingernails and toenails, beyond composting (which Bamboo already covered).
I've personally tried using my own uncomposted hair on plants and in the soil without particular plants before, but I haven't done it enough to have anything much to report (it didn't hurt the plants—and as far as I can tell, the hair did break down, but I haven't been looking for it, particularly; I have tilled the ground where I put it, though, and I didn't see any hair come up).
I tried it in a container once or twice (again, it didn't hurt the plants, but I didn't really notice if it helped or not).
The great thing about hair, though, is that it's probably not going to attract a lot of bugs as it decomposes.