we wrap fresh leafy herbs in kitchen paper and moisten it a bit.
then store either in ziploc but usually in a plastic lidded container to avoid drying out.
the jar of water method to put them in works well if water is changed regularly.
for leafy green herbs like basil: various methods mentioned above.
make it into a puree either freeze as icecubes or mix with oil and put in jar.
laurel leaves: dry and store.
things like parsley, coriander: chop and freeze. keeps quite a while, at least a month
woody herbs like thyme can be dried, so can oregano. rosemary doesn't dry well.