  We study three related homological properties of modules in the BGG category
O for basic classical Lie superalgebras, with specific focus on the general
linear superalgebra. These are the projective dimension, associated variety and
complexity. We demonstrate connections between projective dimension and
singularity of modules and blocks. Similarly we investigate the connection
between complexity and atypicality. This creates concrete tools to describe
singularity and atypicality as homological, and hence categorical, properties
of a block. However, we also demonstrate how two integral blocks in category O
with identical global categorical characteristics of singularity and
atypicality will generally still be inequivalent. This principle implies that
category O for gl(m|n) can contain infinitely many non-equivalent blocks, which
we work out explicitly for gl(3|1). All of this is in sharp contrast with
category O for Lie algebras, but also with the category of finite dimensional
modules for superalgebras. Furthermore we characterise modules with finite
projective dimension to be those with trivial associated variety. We also study
the associated variety of Verma modules. To do this, we also classify the
orbits in the cone of self-commuting odd elements under the action of an even
Borel subgroup.
