We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $\mathcal{ M}(p)$, $p\geq 2$. The first category consists of all finite-length $\mathcal{ M}(p)$-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra $\mathcal{ W}(p)$. We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.
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