We give infinite lists of translation surfaces with no convex presentations. We classify the surfaces in the stratum \({\mathcal{H}(2)}\) which do not have convex presentations, as well as those with no strictly convex presentations. We show that in \({\mathcal{H}(1, 1)}\), all surfaces in the eigenform loci \({\varepsilon_4, \varepsilon_9}\) or \({\varepsilon_{16}}\) have no strictly convex presentation, and that the list of surfaces with no convex presentations in \({\mathcal{H}(1, 1) \smallsetminus (\varepsilon_4 \cup \varepsilon_9 \cup \varepsilon_{16})}\) is finite and consists of square-tiled surfaces. We prove the existence of non-lattice surfaces without strictly convex presentations in all of the strata \({\mathcal{H}^{\mathrm{(hyp)}}(g- 1, g - 1)}\).