We present experimental and theoretical studies of nonlinear dynamics induced by optical feedback in vertical cavity surface emitting lasers (VCSELs) focusing on the impact of polarization mode competition. We unveil the bifurcation mechanism leading to multiple channels of interchanging linearly polarized (LP) steady states, study the way of their destabilization and the role of the external cavity modes and the polarization of light in this process. We distinguish two situations depending on the dichroism present in the VCSEL. In the first one, as typical for semiconductor lasers, with increasing the optical feedback strength the VCSEL exhibits a Hopf bifurcation from steady state to a limit cycle followed by a torus bifurcation to quasiperiodic behavior and then can become chaotic. Bistability between different single polarization dynamics: steady-state, periodic, quasi-periodic and chaotic dynamics is possible, however, the suppressed polarization mode is never excited. In the second case, when the VCSEL is biased in the region of polarization bistability, increasing the feedback strength can not only cause polarization switching but can also lead to chaotic emission in the two LP modes. Polarization switching can happen from a limit cycle in the x-LP mode to a steady state in the y-LP mode which then undergoes consecutive Hopf and torus bifurcations and becomes chaotic. Bistability between chaotic two LP mode emission and periodic single LP mode emission is demonstrated. We also consider the case of polarization selective optical feedback and the influence of the spontaneous emission noise or external noise in the system. For the last situation we demonstrate slow antiphase hopping between the two LP modes together with rapid anticorrelated oscillations in the LP intensities at the external-cavity frequency. These last dynamics can be strongly enhanced by the noise leading to the phenomenon of coherence resonance. Finally, we consider the transition from long to short external cavity and show that the typical low-frequency fluctuation chaotic dynamics is anticipated by a regime of regular dynamics - fast pulses at the delay time modulated by a slow time periodic envelope. We discuss the interplay of optical feedback and polarization dynamics in this regime, too.