The rim is what your fingers feel first in a handmade piece. A slightly rough edge on a tiny saucer, spoon rest, or shallow bowl makes it seem unloved; a too-smooth edge can look thin, wavy, and weak. Smoothing the rim is not a finishing step only; it is an active step in the process of forming a vessel or sculpture, since each movement transforms the clay.
The first temptation for a potter, as the edge becomes uneven, rough, or irregular, is to dampen the finger or dip the sponge to soften it. But this will sometimes soften and thin the rim, leaving a puddle of slip that travels around the clay, sometimes slumping, stretching, and deforming what you have already formed. A piece that has been pulled down and thinned near the rim and is already fragile will be even more so by water.
It takes a bit of patience not to rush for the water, to allow the clay to firm a bit so you can work gently at the rim, especially if you are pinching or throwing it up from a flat slab. The rim should yield to pressure but not be sticky and deforming when you press it, and a barely damp sponge, fingertip, or rounded wooden tool can compress and reshape without saturating it with water.
Go slowly around the rim; turn the piece rather than moving your hand along it, applying light pressure. Too much pressure will flatten or bulge the wall, and if you try to compress the rim, it might be thinner or more pointed. It is more about joining and rounding the clay at the rim rather than making it look perfectly finished.
Making a pinch pot just for this exercise, or a small coil to try on your own, will show you how it works. Try using a water-saturated sponge, a dry sponge, then, once the clay has set some, a wet finger. Let it rest for a moment and see how they feel. The one with the wet sponge will feel smoother and will look smoother, but it will be weaker or stretched; the one that waited longer and has been smoothed will look and feel better because the clay is able to stand up to the pressure.
Do not apply stamps, marks, carving, or underglaze if the rim is already deforming with your finger and you have to work it to keep the rim in place. First step: check the profile and see if the rim rises at the same level across the piece, and then check if it is level on a flat surface to make sure that smoothing didn’t move one part of the rim.
And again, the rim does not have to be perfectly even on a handmade piece. It is more that it looks good, it feels like it has been thought about and it can be handled, it feels strong enough for the next stage of the piece. Stop the moment that the rim is rounded and the wall has thickness, when the clay is free of scratches and rough bits, and do not add extra touches just to make it look smoother because this is how rims get too thin. And it is not always the first touch on a raw piece that causes trouble, but the five more after the rim was already ready that weaken it.