This build is created for my League clan mates, in particular the less-experienced ones.
Since I am maining support I am hoping to teach my build and maybe produce someone in clan who runs a better Morgana than me.

YOU MUST UNDERSTAND THIS BEFORE YOU COMMENT+VOTE ON THIS BUILD.

As though I gave a dung about people's "opinions" anyway. This is really just for my clan and IDFC what you think of my build. (Unless you're in my clan, then that's important :P)

So the law of opportunity cost: Everything is worth the next best alternative.

EXAMPLE.
For 120 minutes spent watching a movie, you could spend that time studying, or play 3 40-minute games of LoL.

That 120 minutes is valuable.
Say (like me) you really don't like movies, and you're too lazy to study. So I choose to play League of Legends. The cost of me playing League for 120 minutes is a 120 minutes of movie OR 120 minutes of studying.

Now, are movies bad? NO, movies are awesome. Is studying bad? NO, it gets me into a better college. So all three alternatives are good. But that doesn't matter. I'm picking what I feel is THE BEST OPTION out of the three.


IN CONCLUSION
When I say the term "inferior" it doesn't mean that what I'm referring to is bad or that you can't win if you build it. If it was bad Riot would take it out of the game.
What I am saying is that there is a better alternative. And who wouldn't settle for better?

Runes
I put almost all my runes into AP and mag pen, pretty self-explanatory. Only in low ELO will you be able to get a full kill under level 6, so I used AP/level, which overtakes flat AP at level 7.

Then my defence runes are HP/level. With my item build, Morgana will boast a lot of armor and magic resist, so the only thing needed is HP, and so I have that.

Masteries
Standard caster 9/0/21. If you don't already have an AP page that looks almost exactly like the above you're probably low ELO.
Another decent option: -1 point in Greed and +1 in Utility Mastery. Works better in high ELO where people actually know how to jungle and give you blue to control lane. FYI the opportunity cost is maybe 1 or 2 HP or mana potions.

Spells
Flash+Ignite standard for all casters nowadays. But in case you're too noob to get it...

-Flash: Enemy has like 100 HP. Throw dark binding, enemy flashes, QQ. Then you see Xia0s0uL flash into an enemy's face and Dark Binding at melee range. QQ moar.
-Ignite: Enemy has like 100 HP, you have 0 mana, QQ. Then you see Xia0s0uL Flash into range, Ignite, and autoattack for the kill. QQ moar.

All other spells are pretty much inferior on Morgana. Flash is the best multipurpose spell and Ignite is best for a medium-high-cooldown caster like Morgana who will need another source of damage (see example above).

Skill Build
People like to max W or alternate between Q and W.
Q increases by 55 damage/level and snare duration by 0.25 seconds/level. W increases by 15 damage/second/level. So if I level Q, A Q+W combo would gain 55 damage + 15*0.25 damage from extended snare + W time. (Also keep in mind the snare duration prevents enemies from escaping Soul Shackles.) If I level W, I only gain 30 damage (level 1 snare is 2 seconds).
By the law of opportunity cost, what's better? Think for yourself.

Start with Doran's Ring. Boots + 3 HP pots can work too but it is very difficult to harass Morgana anyway.

Core:
Doran's Ring x2
Sorcerer's Shoes
Rabadon's Deathcap
Void Staff

Very standard AP core build.

2 early Doran's Rings are the most optimal for scaling into the mid and lategame. Top players tend to go overboard with the Doran's Items and get 3-4 because they lasthit 99% of the creeps in lane anyway. 2 is perfect for the normal players like you and me. Anything less will provide inferior scaling. (I'd show you the lovely econ/math work but it's all a bunch of mumbo-jumbo to most people anyway.)
Keep in mind that the Doran's Ring is the most efficient caster item in the earlygame, so once you hit 6 and get 2 Rings YOU SHOULD GANK. (I gave a gameplay tip, promise broken lol)

Deathcap right after 2x Rings. Rings-Deathcap rush is the scariest caster build out there and a too many players do not understand this.
Huge misconception about APs is that they must build Rod of Ages, and many players just build it first, or even worse, build two Rings and then rush Rod. This is highly inferior on burst-y instant casters like Annie, Gragas, and ofc Morgana. (However, it is the optimal build for more DPS-y casters like Karthus, Anivia, and tanks like Singed or AP Blitzcrank...but that's for other discussions in other builds/guides)

After that, you get Void Staff. Void Staff is actually a pretty questionable core item. If you play a really good Morgana, the game never goes that late and typically very people get significant amounts of Magic Resist.
Some people like to get Abyssal Scepter over Void. In terms of magic resist-lowering, the Abyssal is far inferior to Void Staff. It has more use on tanks like Morde and Singed who balance offensive and defensive stats, but not on Morgana.

Lategame:
Morello's Evil Tome (replaces Doran's Rings)
Zhonya's Hourglass
Banshee's Veil

Morello's Tome real underrated. Some people sell their Doran's Rings lategame and "OMG WHY I ALWAYS HAVE NO MANA?!" You sold your mana regen item, stupid. :P
So for those guys who decide to sell their Rings and find themselves out of mana, Morello's Tome solves the mana problem, and it also gives CDR and AP. It's almost a permanent Blue Buff really.

Banshee's is my defensive item of choice. A lot of people disagree with this and then get pulled by Blitzcrank / Shunpo-ult'd on by Katarina / Amumu bandaged / Galio flash-ult / Ashe arrow / Dark Binding from another Morgana / every other spell you can think off. Admit it, you aren't going to have a perfect Black Shield timing, and Banshee's Veil makes your life a lot easier. Plus the stats are SO SO SO SO GUD.

Morgana as of this patch is freaking OP. So enjoy it while you still can, because Riot doesn't have 1% of the genius of Icefrog and will nerf drastically if they feel the need to. (Can't wait for DotA2 OMGGGGGGGG!!!!)

GLHF