Two stabilizer codes are said to be equivalent if they can be related by non-entangling Cliffords, i.e. by local Cliffords and SWAP gates.
There are unique stabilizer codes for the parameters $ [[2,0,2]], [[4,2,2]], [[5,1,3]], [[6,0,4]] $.
All $ [[2,0,2]] $ stabilizer codes are equivalent to a Bell state with stabilizer generators $ XX,ZZ $.
All $ [[4,2,2]] $ stabilizer codes are equivalent to the "quantum repetition code" with stabilizer generators $ XXXX,ZZZZ $.
All $ [[5,1,3]] $ stabilizer codes are equivalent to the standard $ [[5,1,3]] $ code https://en.wikipedia.org/wiki/Five-qubit_error_correcting_code, see Equivalence between Quantum Error Correcting codes and uniqueness of the $[\![5,1,3]\!]$ code.
All $ [[6,0,4]] $ stabilizer codes are equivalent to the quantum hexacode (the stabilizer code corresponding to the classical hexacode over $ GF(4) $ https://en.wikipedia.org/wiki/Hexacode )
Are there any other parameters for which there is a unique stabilizer code?
