Microswimmers (planktonic microorganisms or artificial active particles) immersed in a fluid interact with the ambient flow, altering their trajectories. By modelling anisotropic microswimmers as spheroidal bodies with an intrinsic swimming velocity that supplements advection and reorientation by the flow, we investigate how shape and swimming affect the trajectories of microswimmers in surface gravity waves. The coupling between flow-induced reorientations and swimming introduces a shape dependency to the vertical transport. We show that each trajectory is bounded by critical planes in the position-orientation phase space that depend only on the shape. We also give explicit solutions to these trajectories and determine whether microswimmers that begin within the water column eventually hit the free surface. We find that it is possible for microswimmers to be initially swimming downwards, but to recover and head back to the surface. For microswimmers that are initially randomly oriented, the fraction that hit the free surface is a strong function of shape and starting depth, and a weak function of swimming speed.
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