I know in the Clone Wars TV show, there are numerous examples of Jedi and Separatist leaders being taken prisoner. Is there any instance where either the Separatists took regular clone troopers prisoner, or where the Republic took battle droids prisoner? Or was it simply a fight to the death?
In the clone wars TV show there are a few episodes where clones were taken prisoners but always only in combination with high ranking officers. Although it seemed that often the droids had "take no prisoners" order it seems like the clones were time and again taken as prisoners. Although for how long is questionable. In one of those episodes the clones would be executed one by one if the jedi don't reveal crucial infos. In others they were freed just mere minutes later. So it is not really known if they are just interrogated and then executed or kept as prisoners if they are no longer useful.
On the other hand droids even though they seem to be able to at least have some individual thoughts are still unable to go against their programming which is to follow orders. Thus without any orders to give up they won't give up, but if that order is given they give it immediately. This happened a few times during the course of the series, and as there is one episode where they use a group of reprogrammed battledroids against the separatists it seems likely that even if most are dismantled / destroyed they still have a few that are still functional and are just reprogrammed. Aside from that they will just dismantle droids and get their memory cores out to see if they have anything useful stored there (as even the jedi don't seem to bat an eyelash at such actions).
Prisoners of the Republic
In the Season 7 episode "Unfinished Business", Mace Windu attempts to take a horde of battle droids prisoner:
MACE WINDU: My name is General Mace Windu of the Jedi Order.
At this point of the Clone War, I have dismantled and destroyed over 100,000 of you Type 1 battle droids.
I'm giving you an opportunity to peacefully lay down your weapons, so that you may be reprogrammed to serve a better purpose than spreading the mindless violence and chaos which you have inflicted upon the galaxy.
BATTLE DROID: Blast them! [opens fire]
OBI-WAN: Well, I guess it was worth a try!
Prisoners of the CIS
Its canon status is unclear, but in cutscenes in Battlefront II matches where the Separatist faction beats the Republic faction, the droids are often shown taking clones prisoner: