This study deals with the asymptotic performance of a multiple-spur cancellation scheme. Radio frequency transceivers are now multi-standard and specific impairment can occur. The clock harmonics, called spurs, can leak into the signal band of the reception stage, and thus degrade the performance. The performance of a fully digital approach is presented here. A one-spur cancellation scheme is first described, for which we exploit the a priori knowledge of the spur frequency to create a reference of the polluting tone with the same frequency. A least-mean-square (LMS) algorithm block that uses this reference to mitigate the polluter is designed. However, due to imperfections in the physical components, there is a shift between the a priori frequency and the actual frequency of the spur, and the spur is affected by Brownian phase noise. Under these circumstances, we study the asymptotic and transient performance of the algorithm. We next improve the transient performance by adding a previously proposed adaptive-step-size process. In a second part of this paper, we present a multiple-spur parallel approach that is based on the one-spur cancellation scheme, for which we provide a closed-form expression of the asymptotic signal-plus-noise interference ratio in the presence of frequency shifts and phase noise.