how can I dispose of a 19" computer monitor, non- working?

Answer:

A 19 might be worth fixing if it's not over 6 years ancient,

or it might have a trade contained by value. Check for Dutch auction coupons.
the dump
A trash can.
it depends what state your in. some states you can newly throw it away, others you have to hold it recycled. find out before dumping it. here are websites where you can find out nearly recycling computer parts. below are some examples. look for "dispose computer monitors" on your favorite survey engine.