In this paper we report on a theoretical investigation of the nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers (VCSELs) under current modulation. Special attention is given to the comparison with a previously studied case of single-transverse mode VCSEL emitting in two orthogonal polarizations. The consideration of spatial effects in VCSEL modifies the polarization dynamics that accompanies the period doubling route to chaos for large modulation amplitudes. Depending on the modulation parameters, the excitation of a higher order transverse mode may either induce chaotic pulsing in an otherwise regularly pulsating VCSEL, or induce a time-periodic pulsing dynamics in an otherwise chaotic VCSEL. Bifurcation diagrams obtained for different modulation frequencies, several values of the dichroism, and different transverse mode characteristics allow us to identify the different scenarios of polarization dynamics in a directly modulated VCSEL. Temporal analysis of carrier number radial profile reveals considerable changes for the multitransverse mode case only constituting the physical origin of the reported changes in the temporal and polarization dynamics.