  In this paper we make an analysis of self-similar solutions for the mean
curvature flow (MCF) by surfaces of revolution and ruled surfaces in
$\mathbb{R}^{3}$. We prove that self-similar solutions of the MCF by
non-cylindrival surfaces and conical surfaces in $\mathbb{R}^{3}$ are trivial.
Moreover, we characterize the self-similar solutions of the MCF by surfaces of
revolutions under a homothetic helicoidal motion in $\mathbb{R}^{3}$ in terms
of the curvature of the generating curve. Finally, we characterize the
self-similar solutions for the MCF by cylindrical surfaces under a homothetic
helicoidal motion in $\mathbb{R}^3$. Explicit families of exact solutions for
the MCF by cylindrical surfaces in $\mathbb{R}^{3}$ are also given.
