  Continuing recent efforts in extending the classical singularity theorems of
General Relativity to low regularity metrics, we give a complete proof of both
the Hawking and the Penrose singularity theorem for $C^1$-Lorentzian metrics -
a regularity where one still has existence but not uniqueness for solutions of
the geodesic equation. The proofs make use of careful estimates of the
curvature of approximating smooth metrics and certain stability properties of
long existence times for causal geodesics. On the way we also prove that for
globally hyperbolic spacetimes with a $C^1$-metric causal geodesic completeness
is $C^1$-fine stable. This improves a similar older stability result of Beem
and Ehrlich where they also used the $C^1$-fine topology to measure closeness
but still required smoothness of all metrics. Lastly, we include a brief
appendix where we use some of the same techniques in the Riemannian case to
give a proof of the classical Myers Theorem for $C^1$-metrics.
