I sew Scout patches on with a Singer 538 Stylist using a Singer 2020 Universal 80/11 needle and Coats & Clark Dual Duty Plus or XP thread. Using the straight stitch setting, I sew in the satin stitch edge with matching color thread on top, bobbin thread color to match the garment for underneath, and with a stitch length about midway on the machine (2 out of 4). I sew them on cold and appreciate new unused patches. I have the most problems sewing patches that have been ironed on but didn’t stick well. (Maybe the heat from the dryer makes them different?)
I should point out that the 538 was designed with fiber gears, and I have replaced one of mine in the 6 years that I have been using it. Gear failure was from age of the gear, and happened while hemming a jeans leg.
I use the smallest needle that I can for the thread size being used, especially on my home-use type machines. The larger the needle, the more machine effort it takes to pierce the material.
My wife sews Scout patches on the same way, except that she uses a Singer 401A.
CD in Oklahoma