I didnt really read what the other guys said so I'll just give you a little overview.
First of all, which sounds pretty basic, well which actually is pretty basic, you should be in a good position the entire time, one wrong step could basically lose you the entire lane and make your late game a pain.
Try to stay behind the creeps all the time, also you should avoid standing in brushes which you are not 100% about that they are not warded. When you know her Q is on cooldown (11 seconds on all levels) try to put a little pressure, but only if you have vision on river and tribrush.
You can try to put her in sticky situations with forcing her to use her E (23/21/19/17/15 seconds cooldown) when she used it you just gonna wait for you stuff to come back up since its most likely on a lower cooldown then her E and just engage on their ADC.
Post level 6 though, you should try to stay out of her ult range (600 I think not too sure) or atleast in the outer area.
I think I covered the most important stuff. Possible picks against her are Caitlyn, Ezreal, Tristana, not just because they all have an escape with lower cd than Moragan's Q but also because they can attack out of high range.
When it comes to supports I cant provide any personal experience but according to statistics Braum and Karma are the only supports who win more than they lose against Morgana.
I hope I could help you with that lil overview, when you liked it gimme a thanks mate
Good luck on the Rift!
