We will introduce a local notion of Hilbert-Schmidt stability (HS-stability), partially motivated by the recent introduction of local permutation stability by Bradford. We will discuss some basic properties, and then establish a local character criterion for local HS-stability of amenable groups, by analogy with the character criterion for HS-stability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HS-stability. Finally, we show that infinite sofic (resp. hyperlinear) property (T) groups are never locally permutation (resp. HS-) stable, answering a question by Lubotzky. This is based on ongoing joint work with Francesco Fournier-Facio and Maria Gerasimova.