I have an assortment of very basic canvas tote bags which are patterned, but not with a logo. One is leopard spots, one is red with white polka dots, one has an American flag print, etc. If I'm going out shopping and I remember to grab one they are very handy.
I also have a faux Longchamp Le Pliage style bag that I often carry if I'm going out shopping. I'll just take the bare minimum of "purse" items (e.g. sunglasses, lip balm, wallet) so that my bag is mostly empty.