  We consider the typical behaviour of random dynamical systems of
order-preserving interval homeomorphisms with a positive Lyapunov exponent
condition at the endpoints. Our study removes any requirement for continuous
differentiability save the existence of finite derivatives of the
homeomorphisms at the endpoints of the interval. We construct a suitable Baire
space structure for this class of systems. Generically within our Baire space,
we show that the stationary measure is singular with respect to the Lebesgue
measure, but has full support on $[0,1]$. This provides an answer to a question
raised by Alsed\`a and Misiurewicz.
