  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.
