Aloha :)
In my eyes there are enough guides for Morgana and all playstyles are explained, but I think ppl dont take enough note of the need for edit and changes in case you want to play 3on3.
And as Morgana is my main-character for 3on3 I felt like sharing some thoughts with you and hope for feedback.

There is a world far away populated by graceful and beautiful winged beings gifted with immortality, where an ancient conflict still rages. Like so many conflicts, this war split families. One side proclaimed themselves as beings of perfect order and justice, fighting to unite the world under their law and strong central governance. Those that fought against them saw their kin as tyrants, creatures incapable of seeing the larger view, who would sacrifice individuality and freedom for the illusion of efficiency and safety. Morgana was one who fought against what she perceived as the tyranny of her kind, and for that she was branded ''fallen.'' Morgana was not innocent, having plumbed forgotten ways to gather forbidden might to become a powerful mistress of the black arts. This goal was driven by her obsession to defeat the general of the opposition's army - her sister, Kayle.

While the two were in fact birth-sisters, Kayle struck the first blow by disowning any filial connection when Morgana refused to join her cause. Eventually, Morgana grew in power enough to not only reach, but challenge Kayle. As the time approached when the two would meet in what could be their final conflict, Morgana was suddenly summoned to Valoran. At first, Morgana made a deal with the League's summoners to fight in exchange for greater power. With the advent of Kayle into the League, Morgana now willingly fights in the League of Legends for the privilege of being able to destroy her sister again, and again, and again. She lies in wait for the day the bonds of the Institute of War no longer hold her, and on that day she plans to destroy Kayle once and for all and return home a hero.

Some thoughts to the Summoner Spells:

I think those are the best. For 3on3.
Ghost:
In my eyes better for 3on3 than flash, because ways are not that long and during Ghost you always can make it back to the tower. So this is a great spell to escape in early and midgame and as 3on3-games dont take that long you wont really have a lategame.
It also compliments your ultimate VERY well. Even better than Flash. You will almost always be able to proper place your ultimate AND keep up. It also gives you a bigger control of your movement.

Ignite:
Awesome for early game and gives you that edge you need to kill the enemy after you harrassed him/her down.
Also you should never forget that it blocks healing and regeneration for 50(!)%. And as healing is a little overpowered in 3on3 (usually you can just overwhelm the enemy and just nuke him down but in 3on3 its kinda different because you are only 3, who wouldve thought that ;D ) and this way you can even 1on1 a Mundo 8).

Note: In case you dont like Ghost for some reason, I suggest you to go for Clairvoyance or Flash.

As you may have noticed, I set up 3 different builds for you. In the following i will explain what those are for.

First and second build:
Those builds are for laning bot with a teammate.
First Items are Sapphire Crystal and potions. Only for beeing able to stay in the lane longer.
With her Tormented Soil and those items you should be able to farm pretty long and enough gold to go back and buy boots and in the best case the full catalyst the protector.
If a fight comes up, you should be sure to hit them a few times with tormented soil while farming BEFORE the actual fight. Then you have your flat-ap-runes, ignite and their lowered magic resistance due to tormented soil and trust me, this is a very powerful combination.
After the next recall you go for t2 boots and then try to finish rod of ages as soon as possible as it gets stronger over time and you want to be strong the soonest ;)
so this is your mainbuild: Ionian Boots of Lucidity and Rod of Ages. After this you should get a Sight Ward and place it at the red buff. Usually i would advice you to buy it earlier, but you need the money for the rod of ages.
Now FINALLY to the Differences in the first and second build:
In case you face an AD-heavy team, you want to get Zhonya's Hourglass, in case you face an AP-heavy team, you want to get Abyssal Scepter. If you have a team with a good combination of both, you should go for Zhonya's Hourglass or if you also have a tank just skip these items and continue with Rabadon's Deathcap
. In the late game (you probably wont come this far just because the game is over before) you want to decide between Lich Bane and Will of the Ancients
- the latter in case you have an AP-heavy team and there arent much towers to push left.

The 3rd Build:
This build is for soloing top. As you want to be more durable and still have some mana-reg to stay in lane, you go for Doran's Ring. In this build you get the Sight Ward earlier, because Morgana is a great pusher and it will be easy for you to harrass the enemy tower. This also means you are vulnerable to ganks so you want to place that ward as soon as possible. In this build you will have to decide for either Abyssal Scepter, Zhonya's Hourglass or Rabadon's Deathcap as your third item and Lich Bane or Will of the Ancients as your last item too.

I suggest not to replace Rabadon's Deathcap, as it fits to your damage-output and brings you to OMG-SHE-IS-FREAKIN-OP status and it also boosts your shield vastly (and with it your possibility to support).

In case you face a lot of offtanks or champs that build MR, Void Staff is definitly also worth thinking about. Thanks to just1rape for his constructive feedback.

First of all, I wanna say something to my skill order:

You take Dark Binding first, because you want to get first blood and this is pretty much the best spell (besides ignite) to get you that in a lvl 1 fight.
Besides that, this is situational. In case you wont for sure risk a fight before lvl 2 then you should skill your Tormented Soil first as this is your farm and harass spell.

Then you skill your Black Shield, because nobody will be able to CC you after that so you can always stay in the back and farm and harass. To be able to do that you skill Tormented Soil to lvl 3 before becoming lvl 6 because together with your runes you will be able to kill the 3 caster minions only with this one spell, allowing you to get a great and safe farm.

For mid- and late-game you max your Dark Binding first as it helps you to catch single targets for your team and has a really great dmg-output. Then you max Tormented Soil as the combinations of catching a target and placing this spell under it will be your main damage-output.
You can also think about a different skill-sequence where you give your Black Shield some more skill points early on if your team has nobody to tank and needs support or the enemy team exists only of ap- / cc-champs.
Now to your ultimate Soul Shackles.
You CAN use it for single targets. But you should only do this in early game. Usually you want to keep it up for a teamfight. Now to the actual teamfight:
You always want to initiate with your ultimate as soon as all three targets are in range! so what you wanna do, you try to catch the carry with your cage and place your AoE so it will get most possible targets. then you go in close range and use your ultimate. this is when your teammates should go in melee-range (if they are melees) and in general start the teamfight! then pop the active of Zhonya's Hourglass when you get damage and wait for your teammates to do the rest. after your invincibillity you want to place another AoE and get out again to catch the remaining enemies.

With these Builds you will have survivability, damage and the possibility to support. Or in more general words: you are the perfect champ for 3on3 :) at least in my eyes :D

And just to clearify this: you will probably never come over your first Elixier of Brilliance, just because 3on3-matches dont take that long. But in case you will get fed really much and can farm greatly, you will like those upcoming items.

So with these Builds you stay long enough in your lane in early game to farm really good and you will constantly become stronger in midgame (due to rod of ages), which is what matters in 3on3.