The stem of the menstrual cup should not protrude from the vulva. If it does, it will not be comfortable. If this is the case, consider trimming the stem.
If you choose to keep the stem, it is most comfortable to position it vertically.
If you feel any irritation or discomfort from the stem, consider removing it. The purpose of the stem is simply to help you reach the bottom of the cup but not for actual removal. Never pull on the stem to remove — it will hurt and make a mess. Proper cup removal happens only by squeezing the bottom of the cup itself to release the suction. If you can reach the bottom of the cup, then there is really no need for the stem.
Just use sharp scissors and make sure you don't trim too close to the cup itself. Otherwise you will end up with a funnel instead of a menstrual cup! If you are not sure how much to trim, start by trimming only a little at a time until you reach a length that you cannot feel. You can also bevel the edge by making small snips to eliminate sharp edges.