We've stopped using plastic single-use bags down here for over a year and the sky hasn't fallen down.
If fact, I didn't hear a single person complaining about it.
Another important aspect to consider, apart from the waste that these bags generate, is the problems that arise from manufacturing them.
Quickly looking online, I found this...
"Worldwide, a trillion single-use plastic bags are used each year, nearly 2 million each minute. The amount of energy required to make 12 plastic shopping bags could drive a car for a mile." www.earth-policy.org