I can't say I'm the best Morgana support out there but I find her really fun to play compared to other supports. I usually play her mid but I just thought eh why not and i tried her support and she's just amazing. The pressure you put the enemy carries under is higher than a lot of other supports. Also, this is my first post ever so I don't really know how to do a lot of cool things like link pictures or ease of accessibility.

Pros:
Long range on her Q
2-3 sec binding
1.5 sec stun
Her E stops other aggressive supports and possible ganks
Has good damage
Hard to chase(q,e)

Cons:
Timing on E can be hard for beginners
Q can be quite difficult to land
Can't heal carry besides summoner spell

Her q has a really long range and is spammable due to its low cool-down and fair mana cost. You can use this to harass the enemy's ad carry or prevent their support from acting aggressive. It can also be used to punish them if they get too aggressive or out of position. You won't be able to finish off an enemy all the time if they get caught in it, but it is your teams call whether to go for it or not. I don't max q immediately because it's cost increases as you level up but if you're running mana regen runes, it would be a good idea to max q right away

Her w is okay to farm when your teams not in the lane but it doesn't do enough damage to max first or second. It doesn't bring support like her other skills so i take one point on it at lvl 5 just to do a little damage or get farm under the turret.

Her e stops any cc or magic damage going towards whoever she casts it upon. This can stop a lot of jungler's ganking abilities such as Skarner's ult and Nocturne's fear, etc. This also can help your ad carry if say their taric flashes to stun.

Her ult can set up ganks, catch fleeing enemies, or prevent heavy aggression. Late game this just screws up the enemy because they don't really want to target you first because you do no damage compared to the rest of your team(Hopefully). Combined with flash and Zhonya's Hourglass it is really hard for the enemy to stop you and easy for you to stop them.

I run magic pen red, armor yellow, cdr blues and gp10 quints because it's what i use for most supports.

Reds:

Magic pen - gives you decent damage.
Armor - For survivability, but I don't really like it that much because i usually just stay back and spew q's. I have enough armor from the yellows.

Yellows:

Armor - great for survivability, even if you don't get focused, it's always good to have armor.
Health per level - Health is always good to have.
Mana regen - I find that i don't really run low on mana that much but it always more spewing so go ahead if you'd want to.

Blues:

CDR - More spewing!
Ability power/lvl - I run these on ap carries, it works fine on support but not as good as others. You don't need a lot of damage.
Magic Resist per level - If the enemy has heavy ap.
CDR per level - Not exactly recommended because you'll have enough cdr late game but it's cheaper than the CDR runes.

Quints:
Health/Armor - Survivability.
GP10 - This allows you to have 5 gold per 10 at level 1 combined with masteries, so you have an extra gp5 item! More gold = more wards.
Movement - Get to places faster, run away faster, why not? Its quite expensive though, so if you have ip trouble, get avarice instead.

i run 9/7/14 to get some cdr, survivability, and the gold. There are other paths such as 9/0/21, and 0/15/15.

Flash: allows you to escape if necessary, and allows you to flash in the middle of the enemy to use your ult. Must have.

Heal: Makes up for the lack of heal.

IF YOUR AD CARRY IS RUNNING HEAL (or life-steal runes)

Exhaust/Ignite: For aggression.
Clairvoyance: To keep up with your enemy jungler, prevent ganks, set up ganks.

CORE:
I start with the basic support start - Faerie Charm, 3 hp pots, 3 wards. You can start with Faerie Charm, 1 hp pot, 2 green wards, and 1 pink ward if you're confident you won't get hurt much (which i don't ever in solo ques). I usually only go back when I have enough for philo, boots1, and a ward or two, but if I go back with say 700 gold only, I get philo stone and wards instead of rushing boots, so you can start your gp5 early, but if you're struggling in your lane for whatever reason, you can rush boots first. I get Heart of Gold next because it gives you survivability and more gp5. If the game is still around the 15:00 mark or earlier, and you have enough for Kage's Pick, get it, if not don't even bother.

WARDS! As a support your main duty is to ward up the map. Look up ward placements for more detail but as for early game:
Blue Team: Wards up slightly below drag area (against the corner of the wall facing bot lane)
Purple Team: Usually would be enough just warding tri-bush. If your enemy ganks mid and goes downwards, just take caution from there.

Oracles: Get this only when you know you wont lose it soon. Like when you have boots of mobility, good surviability items, or great teammates.

Boots: Depends on what you need.
Mobility: If the enemy is low on CC, you're ahead, and you need speed: get Mobility. Not only does this allow you to get to your team faster, it allows you to go around clearing wards and setting up wards;
Lucidity: If you want more CDR, but again your other items should give you enough CDR combined with masteries and runes.
Merc Treads: If the enemy has heavy aoe cc. (Mumu, Fids, Sona, etc)
Ninja Tabi: If enemy has heavy ad damage heroes.

After the Core and the Boots:
Shurelyas: It helps to get Shurelya's before or after Zhonya's. The movement buff too good.
Zhonya's: It's just amazing on Morgana.
Frozen Heart: CD, stops ad carries, when you ult in you'll have them all with that debuff.
Aegis: It's aura is pretty helpful to your team. Ask your team if anyone is getting it before you get it. If someone else has it, don't bother.

You don't really need any more items than this because you need the last spot open for wards always.

But if you need another item, some great ones are:
Rabaddons: If you want to deal more damage.
Rylais: Aoe Slow, survivability, some more damage.
Abyss Scepter: Damage, reduces enemy MR, gives you MR.
Randiuns: If their ad damage is still too much, get this instead of selling Heart of Gold. (You can now go for that new item that HoG goes into too, give your team some actual ad damage negate.

Draft picks:
If your enemy has early stun supports like Leona, Blitzcrank, Taric, etc. Morgana just counters them really hard.

If your enemy has troublesome cc junglers that can screw up your ad carry, like Udyr, Nocturne, Skarner, etc.

Blind picks:

When you KNOW your ad carry can play passive aggressive. If they play too aggressive, they'll lose too much hp, which you can't help them regain.

Just to have fun. =p

Morgana is a lot of fun to play, but the timings, like all supports, can be quite difficult to master. I hope this helped someone at least xD if it even gets read.