© 2020 Society for Industrial and Applied Mathematics. An initial-boundary value problem with a Caputo time derivative of fractional order \alpha \in (0, 1) is considered, solutions of which typically exhibit a singular behavior at an initial time. For this problem, we give a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. L1-type and Alikhanov-type discretization in time are considered. In particular, those results imply that milder (compared to the optimal) grading yields optimal convergence rates in positive time. Semidiscretizations in time and full discretizations are addressed. The theoretical findings are illustrated by numerical experiments.