  Hawking's singularity theorem concerns matter obeying the strong energy
condition (SEC), which means that all observers experience a non-negative
effective energy density (EED). The SEC ensures the timelike convergence
property. However, for both classical and quantum fields, violations of the SEC
can be observed even in the simplest of cases, like the Klein-Gordon field.
Therefore there is a need to develop theorems with weaker restrictions, namely
energy conditions averaged over an entire geodesic and weighted local averages
of energy densities such as quantum energy inequalities (QEIs). We present
lower bounds of the EED for both classical and quantum scalar fields allowing
nonzero mass and nonminimal coupling to the scalar curvature. In the quantum
case these bounds take the form of a set of state-dependent QEIs valid for the
class of Hadamard states. We also discuss how these lower bounds are applied to
prove Hawking-type singularity theorems asserting that, along with sufficient
initial contraction, the spacetime is future timelike geodesically incomplete.
