McDuff showed that every connected homotopy type can be realized as the classifying space of a discrete monoid, but the monoid she constructs has lots of idempotents.
Question: Which homotopy types are realized as the classifying space of a right-cancellative discrete monoid?
In the commutative case, my guess would be that $BM \simeq B(M[M^{-1}])$, so that the classifying space is aspherical. But I'm less confident that this happens in the noncommutative case.
By universal properties, we have that $BM$ is the classifying space of the homotopy localization $B(M[M^{-1}]^h)$. Thus $BM$ is aspherical if and only if the homotopy localization is discrete.
Further, Dwyer-Kan showed that if $(M,W)$ admits a calculus of fractions, then the homotopy localization agrees with the ordinary localization. When $M$ is cancellative, $(M,M)$ admits a calculus of fractions if and only if it satisfies the Ore condition: $$\forall m_1, m_2 \in M, \exists n_1, n_2 \in M, ~ n_1 m_1 = n_2 m_2.$$ So in this case $M[M^{-1}]^h \simeq M[M^{-1}]$ and the homotopy localization is aspherical. In general, I do not know what happens.
