Typically I start by removing any covers that can be removed, spray in some Clean-Streak, isopropyl alcohol, or other degreaser into the shifter, find the stuck pawls and move them back and forth till they start working properly, use an air gun to spray out the degreaser and bits of old grease, then relubricate with Tri-Flow or Dri-Slide. Gets most shifters working fine again.
There are a few models that might not be worth the time, either because the above method hasn't had a great success rate on them, like 7 speed Deore LX M560, and any of the early 90's models that have two tiny finger-like pawls right next to each other, or because they work so poorly even when they aren't gummed up, such as Acera X ST-M290 or Nexave ST-T300.