  We solve the dynamic equations of motion for a growing semi-flexible polymer,
or fiber, approaching a fluctuating membrane at an angle. At late times we find
three different regimes: fiber {\em stalling}, when fiber growth stops due to
membrane resistence, {\em run-away}, in which the polymer bends away from the
membrane, and another regime in which the membrane response is nonlinear and
tubular membrane {\em spicules} are formed. We discuss which regions of the
resulting `phase diagram' are explored by (i) single and bundled actin fibers
in living cells, (ii) sickle hemoglobin fibers in red blood cells, and (iii)
microtubules growing within artificial vesicles. We complement our analysis
with full 3-dimensional stochastic simulations.
