Summary: Spurious numerical solutions to the steady Burgers equation are studied. Results on the existence and uniqueness of approximate solutions given by R. B. Kellogg, G. R. Shubin and A. B. Stephens [SIAM J. Numer. Anal. 17, 733-739 (1980; Zbl 0463.76069)] are extended to consider linear finite elements with a non-uniform mesh, a source term or upwinding. Under specific assumptions the nonlinear effects of mesh adapting, upwinding and source term are discussed. It is shown that mesh adapting does not necessarily improves the situation concerning symmetry breaking bifurcations, and that for a sufficient amount of upwinding such bifurcations can be prevented. By linearized analysis, the local existence of spurious solutions bifurcating from constant solutions is proved. As a complement to some of the theoretical developments, global bifurcation diagrams are presented. They were obtained numerically by means of the arc-length continuation method.