Like many, I keep the trash in a small pail with a lid and use grocery bags. My pail is an old 4 litre that I used to buy my ice cream in. We quit when the kids left home. Yikes, that was 25 years ago. We're down to the last two pails and I won't be buying ice cream like that anymore. Storage is under the sink.
If you don't have a space in a cupboard, consider a small lidded pail so there won't be mishaps if you stop quickly.
Not for garbage, just extra storage, I tucked a medium sized pail along side the toilet. It has a lid and keeps things from rolling around.