  In this work, we study the Aharonov-Casher effect in the $(2+1)$-dimensional
Dirac oscillator coupled to an external electromagnetic field. We set up our
system in two different scenarios: in the Minkowski spacetime and the cosmic
string spacetime. In both cases, we solve exactly the Dirac oscillator and we
determine the energy spectrum and the eigenfunctions for the bound states. We
verify that in the Minkowski spacetime, the Dirac oscillator spectrum depends
linearly on the strength of the magnetic field $B$, and on the Aharonov-Casher
phase. In addition, we explicitly obtain the corrections on the Dirac spinors
and the energy levels due to the curvature effects in the cosmic string
background. Finally, we investigate the nonrelativistic limit and compare our
results with those found in the literature.
