Okay, so I finally got a mission with some biomass in it. After having been asked to acquire some for quite some time (research is at a standstill because of this).
Problem is: how do I actually get a sample? I can't seem to get a sample before or after I blow it up and it neither appears in the list of things I've acquired from the mission, nor in the R&D screen. Is there some special technique I'm missing or is it a bug of some kind?

I assumed you get a biomass sample whenever you do a mission in the biomass but it may not always be case. Some people have apparently been unable to get a sample for the entirety of their game. Just keep plugging away. It can be frustrating if you don't get another mission in the biomass though.
Most people tend to get stuck with making Biomass repulsors. The details on this can be found here:
http://forum.ufo-aftermath.com/index.php?showtopic=793

It is kind of random if you will or will not get the samples - probably the most samples of everything you will need until eternity are found on those... damn how are those ships called... Well the ones which are used to spread the biomass :D

The word you are looking for I believe is: Planters

Well it seems to have worked now. Thanks guys!

Yeah, those are the ones, thanks! I actually have complete UFO trilogy without DRM, downside is it's completely localized into Czech, so I can't really speak to anyone about it :D

This is why so many enemies in games have very generic names. It makes translation easier. 'Planters' is a pretty good example of this.