Happy with this solution to stockinette scarf curlilng!
I am working with a bulky alpaca wool yarn and had pretty good luck with this solution to my stockinette scarf sides curling:
Purl 1st 2 rows after cast on to keep end from curling.
To keep sides from curling: P1, K1, P1, K1 on the first and last 4 stitches of each row.
Then seed stitch these first and last 4 stitches of each row (so if a knit is facing you, purl it, and if a purl is facing you knit it.)
I always slip the first stitch of each row of a scarf to get a neat edge without the big loopy sloppiness you get if you don't, so....
Since the first stitch you make on each new row is going to be a purl, begin each row by slipping the 1st stitch (with the yarn in front) purl wise, bring yarn to back and Knit 1, Purl 1, Knit 1. When you get to the end of the row, seed stitch the last 4 stitches (K1,P1,K1,P1)
The result has minimal curl and neat and matching side edges.