I’ve had all kinds of herbs growing on the window sill. Often I buy herbs for a specific meal but don’t use the entire stand, so I just put the pot with the rest on the sill. It either survives or it doesn’t. Quite often it doesn’t. The plants seem to have been forced to a suitable size but don’t have proper roots for long term survival.
The above-ground parts are also often quite weak and floppy and don’t stay upright without the support of the plastic cone around it.

Basil often does well, and tastes good, too. Mint is my new favourite: these two have survived and thrived and even look decorative. (This photo is from before we went on our week-long trip; both plants survived but definitely look quite a bit sadder now.)