We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasistatic and dynamical) shear-stress fluctuations as a function of temperature T and sampling time Delta t. The linear response is characterized using (ensemble-averaged) expectation values of the contributions (time averaged for each shear plane) to the stress-fluctuation relation mu(sf) for the shear modulus and the shear-stress relaxation modulus G(t). Using 100 independent configurations, we pay attention to the respective standard deviations. While the ensemble-averaged modulus mu(sf) (T) decreases continuously with increasing T for all Delta t sampled, its standard deviation delta mu(sf) (T) is nonmonotonic with a striking peak at the glass transition. The question of whether the shear modulus is continuous or has a jump singularity at the glass transition is thus ill posed. Confirming the effective time-translational invariance of our systems, the Delta t dependence of mu(sf) and related quantities can be understood using a weighted integral over G(t).