Is it "burning at/on the stakes", "burning at/on the stake", "burning at/on stake" or "burning at/on stakes"? There's an entry for "burn at the stake" on the dictionary, so I am guessing it's correct, but I also see "burn at the stakes" and "burn on the stakes" on Google, so I am wondering if these other versions are also idiomatic, or they're just errors. Strangely, a lot of these unidiomatic expressions are found in books and articles.
The idiom is "(the witch was) burnt at the stake". It is not uncommon to drop "the", and if there are several stakes then the plural would be used.
Burning "on" the stake is less idiomatic, but still sees plenty of usage.
However if you are looking for the "correct" expression then it is "burnt at the stake".
The fixed phrase for the execution of e.g. witches, etc, is "burning at the stake". "Stake" is singular; there is only one; it is a wooden post to which the victim is tied. "Stakes" (plural) is an error. It is a good idea to use dictionaries to verify meaning and usages, since Google will find wrong as well as correct usages.
burn somebody at the stake
to kill someone by tying the person to a post and burning them