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 Z 2 fractionalization and a conventional superconductor. We note implications for recent measurements of the London penetration depth in BaFe2(As1-xPx)2. © 2013 American Physical Society.