I tend to buy scallions (green onions, as I know them) and cilantro a lot. I usually chop them up and toss them in most food I’m stir-frying around in my wok or in a simple bowl of instant ramen. They’re cheap and plentiful, but unfortunately they go bad quickly.
The solution is just placing your herbs in a jar like flowers in a vase with fresh water and covering with a plastic bag and leaving it in the fridge. Like J. Kenji, I rinse the herbs but mostly for convenience for cooking. It’s nice to just grab them and chop them up knowing they’re already clean. In the case of herbs like cilantro, I’ll take a knife and give a quick trim to the ends of the stems to help in water absorption. I’ll use the produce bag from the market to cover the herbs with mostly again out of convenience.
In comparison to just leaving my scallions and cilantro in bags in my fridge, shelf life is about 4-5 days before things look bad. Using the covered jar method, they still look good after 2 weeks and iffy after 3 weeks. Make sure to check on the water every couple of days and dump out murky water and replace with fresh water.
Also, don’t fall for the 3 bunches of cilantro for $1 trick. I don’t think I’ve ever used a whole bunch no less three of them before they go bad.