State estimation is considered for a class of switching discrete-time linear systems. The switching is assumed to be unknown among the various system modes associated with different known matrices. The proposed scheme relies on the combination of the estimation of the system mode with the application of a Luenberger-like observer whose gain is a function of the estimated mode. In the absence of noises, the estimate of the mode can be chosen among the ones that are consistent with the measurements and the stability of the estimation error is ensured under suitable conditions on the observer gains. Such conditions can be expressed by means of linear matrix inequalities (LMIs). The presence of bounded disturbances is also taken explicitly into consideration. In this situation, a novel method based on a minimum-distance criterion is proposed in order to estimate the system mode. Also in this case the error of the resulting estimator is proved to be exponentially bounded.