  The properties of the space $\A$ of regular connections as a subset of the
space $\Ab$ 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 $\A$ is dense in $\Ab$ or not. Moreover, it is
proven that $\A$ has Ashtekar-Lewandowski measure zero for every nontrivial
structure group and every smoothness category. The analogous results hold for
gauge orbits instead of connections.
