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

Geometric Regularity Results on $B_{\alpha,\beta}^{k}$-Manifolds, I: Affine Connections