Introduction: Morgana is a support character, and anyone who tells you, or plays differently...well, as condon would say, "Is bad and should feel bad." This build can't solo a super minion, can't push lanes and isn't for noobs. Read on if you're still interested.
Runes: Your runes are pretty open, but I currently run:
Red: AP, MPen or Hp.
Yellow: Mana regen/level, but dodge is solid here as well.
Blue: AP or Hp.
Quints: Move Speed, since Morgana's pretty slow, but Hp works here too. Also, 3 gold/10 is something else I've had luck with, since my build is terrible at farming.
With all hp red/blues, it's about 180 at 18, but I don't have the IP for them or the dodge runes, and I played Morgana ALL wrong when I first started out. Quints are another 100sh, but I like move speed, especially early on. You don't need CDR blues, that'll be covered later.
Max out your Q first. It's what you use to snare and run, or snare so your carry can get a kill. You don't need money, you need assists, like 100 of them. A good game for Morgana is 0/(anything under 10)/31978478952. Give your carry the kills. Or the tank. Either way, you don't want them.
Masteries: I run a silly 0/16/14 hybrid deal. Again, no CDR, your items will take care of that.
1 Resistance, 3 Hardiness
1 Nimbleness, 3 Harden Skin
4 Veteran's Scars
This gives you solid early game, with a lot of hp to play with at level 1. That should keep you out of trouble until your build gets going.
1 Teleport, 1 Ghost, 3 Perseverence
If you want, you can pick up 2 in the regen and 3 in Quickness. Again, I play around a lot, so do what you want here.
Black Shield + Ghost will, 90% of the time, get you out of any trouble you're in. Unless they bring out all 5, you shouldn't have a problem getting away, since you can't be CC'd.
Teleport is for getting where you need to be(back to your lane, into the fight, etc.)
Other spells you might enjoy:
Flash: I've never had much luck with this. Ghost lasts longer and with her E, you should be gone before they can get you anyway.
Clarity: It works, and you can fool around with your masteries to get the improved version, but Soul Shroud and yellow runes should keep you up just fine.
Clairvoiyance: I'm never any good at picking where to put it, but as support, it can be really helpful. (But I prefer to drop wards everywhere, read on to learn more!)
Things not to take:
Ignite: You aren't looking to kill anyone.
Exhaust: See above.
Cleanse: You have black shield. Use it.
Smite: Never been a fan, but if you have trouble farming, it may be worth it(make sure to get improved from Offense if you do.)
I'm pretty rigid in the items I make, but not the order. I like to finish games with:
Merc Tread's: These will be your boots 99% of the time. If they have very few or no CC, get Tabi.
Aegis: Yay for team buffs.
Soul Shroud: Yay for team buffs and CDR.
Frozen Heart: Yay for other team debuffs, mana and CDR.
Most of my games, I don't get a chance to finish even these, but Aegis is always last. With Shroud and Heart, you have your 40% CDR, lots of mana, lots of mana regen, and decent hp.
As you go back, try to finish Soul Shroud first, it's the most inconsistent of the items, since the belt takes 1110 gold, and every part of Frozen Heart can be done in small pieces(The most expensive parts are the Chainmail at 700 and the heart for 800).
Once you get Soul Shroud/Frozen Heart finished up, get your Aegis. Now, most people will say that the earlier you can get Aegis, the better. Well, sorry, you need CDR before anything else. More Black Shields and more Dark Bindings(for running or assists), is more win than the little buff that Aegis gives. Besides, both the Shroud and Heart are bigger buffs/debuffs for your team than Aegis.
After you've finished those 4 items, if the game is still going, the rest of your purchases should be wards. Sight if they have no stealth and vision if they do. Drop them everywhere. Any piece of brush your team goes near, drop one. You can buy 6 wards at a time, and you should fill up every time you go back. The map awareness granted by your selfless act will give you a HUGE advantage. You'll be able to avoid ganks and set up your own, and you'll be able to get back before the situation becomes dire. Your team will love you and sing your praises.
Black shield gives immunity to ALL CC, and some magic damage, your carry needs it every time you get into a teamfight.
Dark binding is not for damage. Merc Treads negate 40% of it off the top, and cleanse does too. Use it to stop people chasing you, or to catch a fleeing enemy. Let your carry get the kills, you don't need them.
Don't build AP, for the love of god. 40% of the damage on your Q is negated by a 1200 gold item that most people get anyway. Not to mention they won't sit on your W if they get out of your Q. Also, Cleanse now properly gets rid of Morgana's ult, so they don't take the stun immediately, they just run away, fat and happy.
Morgana's ult is for slowing chasers or fleeing enemies. Most people will run the moment they see that line connecting you, so use it when they're low on health so your carry can kill them.
Criticism is necessary for growth, so let me have it.