Is there any scientific evidence that nootropics improve synaptic plasticity / learning ability? I'm attempting to learn a very specific skill (perfect pitch) and I'm considering nootropic supplements.

While I can find plenty of websites offering these supplements and making big claims, I can't find any rigourous scientific experiments backing these claims.

On the other hand I can't find any rigourous scientific experiments debunking them.

I would imagine such an experiment would not be too hard to set up; for example, performance on a simple "pairs" card game could give a measure of short-term memory.

How might I intelligently go about searching for such experimental results?
I'll have to leave it to someone else to provide real expert advice and references on this point but I do offer the following skeptical points.

Brain Training Doesn't Work

To my knowledge, there is no reliable data whatsoever that says domain general brain-training has any effect. Basically, you can spend as long as you like practice pitch recognition, or anagrams, or any other cognitive task, but the transfer to unpracticed tasks will be minimal - you'll simply improve at the task you've practiced. You cannot increase IQ/Working Memory/Cognitive Capacity in this way. See, for example, Owen et al, 2010. The only benefit accrued from these games is in cases of cognitive decline, such as dementia, Alzheimers, and other neurodegenerative conditions, where training can slow decline.

Nootropic Drugs Don't Work

I've seen no evidence, ever, that shows long term benefits of nootropics, beyond the usual, possibly valid claims about fatty acids found in fish. I can't find references for this right now, unfortunately, but really the onus is on those who claim that these supplements do have an effect to point to unbiased evidence.