In this paper, L1/local discontinuous Galerkin method seeking the numerical solution to the time-fractional Stokes equation is displayed, where the time-fractional derivative is in the sense of Caputo with derivative order $α ∈ (0, 1).$ Although the time-fractional derivative is used, its solution may be smooth since such examples can be easily constructed. In this case, we use the uniform L1 scheme to approach the temporal derivative and use the local discontinuous Galerkin (LDG) method to approximate the spatial derivative. If the solution has a certain weak regularity at the initial time, we use the non-uniform L1 scheme to discretize the time derivative and still apply LDG method to discretizing the spatial derivative. The numerical stability and error analysis for both situations are studied. Numerical experiments are also presented which support the theoretical analysis.