This paper addresses the analysis of oscillator phase-noise effects on the self-interference cancellation capability of full-duplex direct-conversion radio transceivers. Closed-form solutions are derived for the power of the residual self-interference stemming from phase noise in two alternative cases of having either independent oscillators or the same oscillator at the transmitter and receiver chains of the full-duplex transceiver. The results show that phase noise has a severe effect on self-interference cancellation in both of the considered cases, and that by using the common oscillator in upconversion and downconversion results in clearly lower residual self-interference levels. The results also show that it is in general vital to use high quality oscillators in full-duplex transceivers, or have some means for phase noise estimation and mitigation in order to suppress its effects. One of the main findings is that in practical scenarios the subcarrier-wise phase-noise spread of the multipath components of the self-interference channel causes most of the residual phase-noise effect when high amounts of self-interference cancellation is desired.