It's always an interesting challenge giving accessories shape and structure when sewing them out of fabric. I can't claim to have tried every product out there so I hesitate to put specific brand name materials on my patterns. The names of the products seem to change somewhat and you may know of something that works even better.
That being said, people ask me on a regular basis what I use on specific patterns. So here's the lowdown:
Marta Studio Bin Pattern: Pellon Extra Firm Stabilizer ( I have also seen this labeled Extra or Ultra Firm Craft Stabilizer). This is a sew-in product, but you can also use the fusible option for added stability. Do not iron it in, just follow the instructions as written. 16 ounce polyester batting: this is a heavier, thicker batting that when quilted gives you a nice, firm lining. Find it by the roll or ask for a heavier, thicker poly batting.
Sigrid Oilcloth Market Bag Pattern: Same Pellon Extra Firm Craft Stabilizer as Marta or you can also use a stiff piece of buckram.
Eva Sewing Basket Pattern: Pellon Double-Sided Fusible Ultra Firm Stabilizer. This is tha same as the sew-in version, but you can iron fabric to both sides. Very handy for shaping the handle on Eva. Fusible heavy weight interfacing: this is the same stuff you would use on garment construction; used to give thickness to the pockets.
Ilsa & Lilla Handlebar Basket Patterns: Pellon Extra Firm Stabilizer ( I have also seen this labeled Extra or Ultra Firm Craft Stabilizer). This is a sew-in product, but you can also use the fusible option for added stability. Do not iron it in, just follow the instructions as written.
I'm also experimenting with fusible fleece interfacing and am really liking Pellon Thermolam Plus. It's heavier and thicker than some other fusible fleeces and gives accessories some nice structure.