Motivated by experiments on splitting one-dimensional quasicondensates, we study the statistics of the work done by a quantum quench in a bosonic system. We discuss the general features of the probability distribution of the work and focus on its behavior at the lowest energy threshold, which develops an edge singularity. A formal connection between this probability distribution and the critical Casimir effect in thin classical films shows that certain features of the edge singularity are universal as the postquench gap tends to zero. Our results are quantitatively illustrated by an exact calculation for noninteracting bosonic systems. The effects of finite system size, dimensionality, and nonzero initial temperature are discussed in detail.