  The problem of classification of real and complex singularities was initiated
by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t.
right equivalence. The classification of right simple singularities in positive
characteristic was achieved by Greuel and the author in 2014. In the present
paper we classify right unimodal and bimodal singularities in positive
characteristic by giving explicit normal forms. Moreover we completely
determine all possible adjacencies of simple, unimodal and bimodal
singularities. As an application we prove that, for singularities of right
modality at most 2, the $\mu$-constant stratum is smooth and its dimension is
equal to the right modality. In contrast to the complex analytic case, there
are, for any positive characteristic, only finitely many 1-dimensional (resp.
2-dimensional) families of right class of unimodal (resp. bimodal)
singularities. We show that for fixed characteristic $p>0$ of the ground field,
the Milnor number of $f$ satisfies $\mu(f)\leq 4p$, if the right modality of
$f$ is at most 2.
