  We have measured small-amplitude rf penetration depth $\lambda(H,T)$ in pure
and C, Li and (Li+C) doped single crystals of MgB$_2$. The effect of doping on
the critical temperature T$_c$ and on the upper critical field H$_{c2}$ was
found to be in good agreement with previous results. We report the presence of
clear signatures of irreversibility in $\lambda(H,T)$, associated with the peak
effect. Carbon doping enhances the signature and shifts its position on the H-T
phase diagram to higher temperatures. Contrary to this, Li substitution has the
effect of suppressing the peak effect to lower temperatures. Analysis of both
zero field (ZFC) and field cooled (FC) measurements suggests that the
hysteresis associated with the peak effect is due to macroscopic screening
supercurrents, $j$, generated during the magnetic field ramp.
