  We present a general theory of the singularity in the London penetration
depth at symmetry-breaking and topological quantum critical points within a
superconducting phase. While the critical exponents, and ratios of amplitudes
on the two sides of the transition are universal, an overall sign depends upon
the interplay between the critical theory and the underlying Fermi surface. We
determine these features for critical points to spin density wave and nematic
ordering, and for a topological transition between a superconductor with
$\mathbb{Z}_2$ fractionalization and a conventional superconductor. We note
implications for recent measurements of the London penetration depth in
BaFe$_2$(As$_{1-x}$P$_x$)$_2$ (Hashimoto et al., Science 336, 1554 (2012)).
