  Let $(S,H)$ be a general primitively polarized $K3$ surface. We prove the
existence of curves in $|\mathcal O_S(nH)|$ with $A_k$-singularities and
corresponding to regular points of the equisingular deformation locus. Our
result is optimal for $n=1$. As a corollary, we get the existence of elliptic
curves in $|\mathcal O_S(nH)|$ with a cusp and nodes or a simple tacnode and
nodes. We obtain our result by studying the versal deformation family of the
$m$-tacnode. Finally, we give a regularity condition for families of curves
with only $A_k$-singularities in $|\mathcal O_S(nH)|.$
