Let \(1\le p\le 2\) and let \(\Lambda = \{\lambda _n\}_{n\in \mathbb {N}}\subseteq \mathbb {R}\) be an arbitrary subset. We prove that for any \(g\in M^p(\mathbb {R})\) with \(1\le p\le 2\) the system of translates \(\{g(\cdot -\lambda _n)\}_{n\in \mathbb {N}}\) is never an unconditional basis for \(M^q(\mathbb {R})\) for \(p\le q\le p'\) , where \(p'\) is the conjugate exponent of p. Here \(\cdot \) denotes a generic variable. We also prove that for any \(g\in M^p(\mathbb {R})\) with \(1< p\le 2\) the system of translates \(\{g(\cdot -\lambda _n)\}_{n\in \mathbb {N}}\) is never an unconditional Schauder frame for \(M^p(\mathbb {R}).\) Several partial results regarding the existence of unconditional Schauder frames formed by a system of translates in \(M^1(\mathbb {R})\) as well as in \(M^p(\mathbb {R})\) with \(2<p<\infty \) are presented as well. Finally, we prove that \(M^1(\mathbb {R})\) does not admit any Schauder basis formed by a system of translates.
