Summary: The properties of the space \(\mathcal A\) of regular connections as a subset of the space \(\overline{\mathcal A}\) of generalized connections in the Ashtekar framework are studied. For every choice of compact structure group and smoothness category for the paths, it is determined whether \(\mathcal A\) is dense in \(\overline{\mathcal A}\) or not. Moreover, it is proven that \(\mathcal A\) has Ashtekar-Lewandowski measure zero for every non-trivial structure group and every smoothness category. The analogous results hold for gauge orbits instead of connections.