  In this paper we consider existence and multiplicity results concerning
affine connections on $C^{k}$-manifolds $M$ whose coefficients are as regular
as one needs, following the regularity theory introduced in arXiv:1908.04442.
We show that if $M$ admits a $B_{\alpha,\beta}^{k}$-structure, then the
existence of such regular connections can be established in terms of properties
of the structural presheaf $B$. In other words, we propose a solution to the
existence problem in this setting. With regard to the multiplicity problem, we
show that the space of regular affine connections is an affine space of the
space of regular $\operatorname{End}(TM)$-valued 1-forms, and that if two
regular connections are locally additively different, then they are actually
locally different. The existence of a topology in which the space of regular
connections is a nonempty open dense subset of the space of all regular
$\operatorname{End}(TM)$-valued 1-forms is suggested.
