We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding whether, given $p\in \mathbb{N}$, there exists such a constant subsequence along an arithmetic progression of common difference $p$. In the special case of uniformly recurrent automatic sequences we explicitly describe the sets of such $p$ by means of automata.