  We introduce and explore a new concept of evasive subspace with respect to a
collection of subspaces sharing a common dimension, most notably partial
spreads. We show that this concept generalises known notions of subspace
scatteredness and evasiveness. We establish various upper bounds for the
dimension of an evasive subspace with respect to arbitrary partial spreads,
obtaining improvements for the Desarguesian ones. We also establish existence
results for evasive spaces in a non-constructive way, using a graph theory
approach. The upper and lower bounds we derive have a precise interpretation as
bounds for the critical exponent of certain combinatorial geometries. Finally,
we investigate connections between the notion of evasive space we introduce and
the theory of rank-metric codes, obtaining new results on the covering radius
and on the existence of minimal vector rank-metric codes.
