  We show that the topological groups $Diff_{+}^{1}(I)$ and
$Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of
the interval and the circle, respectively, admit finitely generated dense
subgroups. We also investigate the question of genericity (in the sense of
Baire category) of such finite topological generating sets in related groups.
We show that the generic pair of elements in the homeomorphism group
$Homeo_+(I)$ generate a dense subgroup of $Homeo_+(I)$. By contrast, if $M$ is
any compact connected manifold with boundary other than the interval, we
observe that an open dense set of pairs from the associated boundary-fixing
homeomorphism group $Homeo(M,\partial M)$ will generate a discrete subgroup. We
make similar observations for homeomorphism groups of manifolds without
boundary including $\mathbb{S}^1$.
