  We investigate the phenomenology of the Higgs sector of the minimal $B-L$
extension of the Standard Model at a future $e^+e^-$ Linear Collider. We
consider the discovery potential of both a sub-TeV and a multi-TeV machine. We
show that, within such a theoretical scenario, several novel production and
decay channels involving the two physical Higgs states, precluded at the LHC,
could experimentally be accessed at such machines. Amongst these, several Higgs
signatures have very distinctive features with respect to those of other models
with enlarged Higgs sector, as they involve interactions of Higgs bosons
between themselves, with $Z'$ bosons as well as with heavy neutrinos. In
particular, we present the scope of the $Z'$ strahlung process for single and
double Higgs production, the only suitable mechanism enable one to access an
almost decoupled heavy scalar state (therefore outside the LHC range).
