Here is a report that summarizes the reusable grocery bag ban trend that is hitting the US for the past 4 years. The article is written by the reason foundation, a libertarian institute in California. The article summarizes many points developed by a lifecycle assessment on plastic bags, a report developed by the UK's environment agency.
Here is a useful table that details most of the bags available for users. If I had to choose, I would use the cotton bags simply because they are washable, bleach-able and easy to disinfect. All the others don't hold up well in the Washing machine. There is fabric technology that integrates disinfectants for infection control scenarios. That would also be an interesting choice.