  Given a link of a normal surface singularity with its canonical contact
structure, we compare the collection of its Stein fillings to its Milnor
fillings (that is, Milnor fibers of possible smoothings). We prove that, unlike
Stein fillings, Milnor fillings of a given link have bounded topology; for
links of sandwiched singularities, we further establish that there are only
finitely many Milnor fillings. We discuss some other obstructions for a Stein
filling to be represented by a Milnor fiber, and for various types of
singularities, including simple classes like cusps and triangle singularities,
we produce Stein fillings that do not come from Milnor fibers or resolutions.
Meanwhile, we discover that there are many contact 3-manifolds that admit
infinitely many Stein fillings but do not admit arbitrarily large ones.
