We study the nonlinear dynamics of a broad-area VCSEL in solitary continuous-wave operation with a comprehensive investigation of its polarization state, lasing spectra, near-field distributions and temporal dynamics. Fluctuations at the frequency of the birefringence splitting and other frequency components develop in a series of bifurcations. The bifurcations coincide with changes of the transverse lasing modes and/or the polarization state, demonstrating the importance of both the spatial and polarization degrees of freedom for the mode competition. In consequence, the inherent nonlinear dynamics of the broad-area VCSEL is significantly more complex than the dynamics of narrow VCSELs.