Few sewing techniques can be as worrisome as setting in a sleeve smoothly into the armscye (armhole) of a garment. The armscye is the fabric edge to which the top of the sleeve is sewn.
Men's shirt sleeves (and women's) are fairly easy to sew into the shirt when there are no deliberate gathers in the sleeve cap, only a slight amount of easing required. There are two methods for inserting shirt sleeves. One involves sewing the side seams of the shirt first and the sleeve under seam separately and then fitting the resulting sleeve tube into the round armhole. A slightly easier method is to use a flat sewing technique where the sleeve top is sewn to the prior stitched shoulder seam first, then sew the sleeve and side seam all in one continuous motion. The flat method works especially well with sleeves that don't require a lot of easing and for children's clothes where the armhole is small and can be difficult to maneuver under the sewing machine's presser foot, but ultimately, it is a pattern instruction or individual preference.
It is important to ease-stitch first at the top of the sleeve cap between the pattern notches. Ease stitching is like stay-stitching on the seam line but with a longer stitch length than for a regular seam. Stitch on the seam line, and 1/4 inch again inside seam line. Afterwards, pull gently on the stitched thread to slightly gather the sleeve cap so the fabric is no longer flat but has fullness, then use your fingers to softly smooth the fabric back to nearly flat. This will cause the sleeve cap to curve slightly to best fit the top of the shoulder's body curve.
Pin the eased shoulder sleeve into the armhole area, sleeve-side uppermost, right sides together matching center and notched or marked areas, distributing any remaining fullness evenly across the sleeve cap. Best to baste stitch this seam before final stitching. The extra time to baste this seam helps to keep the fabric from shifting as you stitch. Stitch slowly so as not to stitch a pinch of fabric into the sleeve seam. Press the sleeve cap over a pressing ham, or tightly folded towel to retain the curved sleeve top. Remove basting thread. Trim seam allowance.
The resulting stitched seam should be visibly smooth allowing for the sleeve cap to curve gently over the shoulder's edge.
Happy happy, sew inspired.