This paper deals with the existence and regularity at the local time of the multifractional Brownian motion
, i.e, a generalization of the fractional Brownian motion to the case where the Hurst parameter
is a Hölder function of time. Under a mild condition on
, existence, joint continuity, and Hölder regularity of the local time of
are considered. The local Hausdorff dimension of the level set is defined as well.