We examined chain degradation in rheological measurements and the effects of molecular weight distributions (MWDs) on rheological data at steady states for polystyrenes in dioctyl phthalate undergoing flow-induced phase separation. The chain degradation occurred when we observed a strong overshoot in transient stresses. The suppression of overshoot by lowering the concentration and the molecular weights of the samples, and/or by making a step increase in the sheer rate, was effective in preventing the degradation. The most significant difference between the rheological data for samples with broad and narrow MWDs is the existence of plateau in the first normal stress difference in the former. Sheer rate dependences of the stresses of broad MWD samples in the shear rate range before the plateau are somewhat steeper than those of narrow MWD samples.