While these bags are technically recyclable when they are clean, the chance of them being recycled is low. Reusing a plastic bag for trash means it is not single-use!
Anything you use twice is not single use.
Have loads of plastic bags at home? Bring them to the store with you next time and reuse them! If the bags are very sturdy and clean, they may be welcome at bag donation boxes found in some libraries.
While these are welcome in the recycling system, it is generally not advisable to simply recycle them if they are still usable. If, however, the bag is torn and otherwise useless, yes put it in the recycling bin.
A plastic bag folded up is lightweight and easy to carry around. You don't need to pay $$$ to keep a reusable bag handy. A plastic grocery bag is the most compact reusable bag you can get. Keep one handy.
These bags are most commonly made of HDPE (Plastic #2) but may also be made of LDPE (Plastic #4) or non-recyclable plastic like PLA (Polylactic acid - a kind of plastic #7). The see-through bags tend to be LDPE and the opaque ones HDPE.
Although both HDPE and LDPE are technically recyclable, it is hard to sort the material for recycling so tends to have very low value as a waste plastic.