You can absolutely do it with a regular sewing machine, you just need fabric with good recovery and the right approach. The important thing is to use a stretch or ballpoint needle (not a regular needle), because it's specifically made for elastic fabrics and doesn't drop stitches as easily. Sew with a medium stitch length and use a slightly stretchy stitch if you can - lots of Singer machines have that built in, it's often called "Elastic Stitch" or something similar. If your machine doesn't have that, a regular straight stitch works too, just make it a bit longer (around 2.5-3mm), and this is really important: sew slowly and let the machine set the pace, not your hands. With a bit of patience, the seams will hold up fine even without a coverlock.