  It is known that Iterated Function Systems generated by orientation
preserving homeomorphisms of the unit interval admit a unique invariant measure
on $(0,1)$. The setup for this result is the positivity of Lyapunov exponents
at both fixed points and the minimality of the induced action. With the
additional requirement of continuous differentiability of maps on a fixed
neighborhood of $\{0,1\}$, we present a metric in the space of such systems,
which renders it complete. Using then a classical argument (and an alternative
uniqueness proof), we show that almost singular invariant measures are admitted
by systems lying densely in the space. This allows us to construct a residual
set of systems with unique singular stationary distribution. Dichotomy between
singular and absolutely continuous unique measures, is assured by taking a
subspace of systems with absolutely continuous maps; the closure of this
subspace is where the residual set is found. The article is dedicated to the
memory of Professor J\'ozef Myjak.
