The Polyakov quantization of the bosonic string is treated in general homogeneous gauges where the world-sheet metric becomes a dynamical variable. We find that the gauge fixed action is governed by an effective, abelianized BRST (+anti-BRST) algebra which in addition to the ghost number charge contains a generator associated with the Langrange multiplier sector. Whereas in the conformal gauge the latter is trivial, in the general case it acquires an anomaly on a curved world-sheet. We calculate the various world-sheet anomalies in different gauges and finally prove that the conformal anomaly, which determines the critical dimension of the string, as well as the sum of the ghost number and Lagrange multiplier anomalies are gauge independent. Furthermore, we comment upon possible ghost number current anomalies from fixing the κ-symmetry in the covariant quantization of the Green-Schwarz superstring.