They make small cutting mats, check in the paper crafting aisle. They also make small rulers. Press a couple pieces of deconstructed clothing (spritz with spray starch), like 2 or 3 shirt fronts. Then roll the fabric loosely. The fabric, rotary cutter, mat and ruler should all fit in a small tote.
If you really want to go with the scissors, a sandpaper pattern is the way to go. Cut the sandpaper with a pair of crummy scissors dedicated to this task or use a box cutter from the $1 store. You will ruin any blade you use to cut this stuff, but the template will not slip at all.
P.S. What is your avatar a picture of? Every time I see it I wonder...