Early game, Dark Binding into Tormented Soil may burn lots of health from your opponents. Be careful when harassing this way, as the former costs lots of mana, especially when maxed early on.
Single Tormented Soil will deal tremendous total damage to minions and will cause your lane to push. However, since Dark Binding lasts pretty long and Black Shield protects from crowd-control, she is really hard to gank.
Tormented Soils might abuse your lane partner's farming. If willing to poke, use Dark Binding instead, as it costs less mana and will probably damage them more, without pushing the lane.
Morgana'sSoul Shackles might be one of the strongest ultimates, combining high damage with huge scaling and long stun. However it requires her to stick to your enemies, making her vulnerable to attacks. If possible, do not go in first.
Thanks to Morgana's scalings, she becomes really strong in late game, dealing 900 (+ 230% AP) with Dark Binding and Soul Shackles. She can also cast Tormented Soil to further increase damage dealt.
Morgana is great at zoning her enemies, forcing to either leave specified area, hide behind minions or just run away from you.
An alternative might be Bond of Stone, but only if planning to get lots of defensive items. Natural sustain and durability might help her survive the additional damage taken though.
Even if she has a shield, Windspeaker's Blessing is not that good option, since it is used way more rarely than any other shields. It also is not supposed to grant defense, but absorb crowd controls.
Morgana benefits from items that grant her ability power, thanks to great scalings on her abilities. Since she needs to get close to hit her enemies with ultimate, look for items that grant both damage and defense.
A mandatory item for her is Zhonya's Hourglass. It grants ability power, armor to survive a few more attacks and cooldown reduction to cast more spells. But the most important is the stasis active, which allows her to become invulnerable for the time holding the tethers. Note, that you cannot move when under stasis, allowing your enemies to run away freely, possibly breaking the bonds. Use it only when needed.
Another great option is Rod of Ages. It increases her health, allowing to survive longer, increases ability power and mana to overcome the great cost of Tormented Soil. It also scales into late game, where she shines at most.
To grant lots of ability power, Rabadon's Deathcap sounds like a great item. It grants lots of damage to her kit and increases her sniping potential on her Dark Binding. However, since she needs to be close to her enemies when Soul Shackling, it is not recommended to get this one as first item.
Somewhat effective might be Archangel's Staff, thanks to huge base mana pool and it's combination with Rod of Ages. It's shield will also make you harder to kill when in the middle of enemy team. Note, that she cannot stack it that fast, so it's not recommended when you are sure the game won't last long.
Luden's Echo greatly improves the sniping capability with her Dark Binding, increasing it's damage to 400 (+ 100% AP) when charged. It also increases your movement speed, helping to stick to your opponents when Soul Shackling.
Supporting Morgana also benefits from ability power and defense, but since you won't kill that much, you might not find enough gold to afford such items.
For starting item, you might consider Spellthief's Edge to upgrade it to Frost Queen's Claim. It will help you gain gold for new and better items, grants stats you need and also give you great active.
Then forcing into support items, the one dedicated for tanks are better on her, thanks to good base health and defense. Items like Redemption and Knight's Vow suit her very well.
For cooldowns, Frozen Heart is the way to go, if you aren't planning to damage your enemies. It will reduce your opponent's attack speed when Soul Shackling, grant lots of mana and cooldown reduction to increase the frequency of cast Dark Bindings.
Since removing crowd controls is always strong, Mikael's Crucible is a valuable option to all supports and Morgana is not an exception.
Since Black Shield makes her almost impossible to gank, force her to cast it before, or, as a jungler, gank her twice in a quick succession. Thanks to its long cooldown, she should not be able to shield herself again. Be wary of Dark Binding's though.
When Morgana is off her Dark Binding, she becomes much easier to poke. Use this time to execute own skills or just hit her with few basic attacks.
Avoiding mentioned ability will also greatly reduce her effectiveness in the next few seconds, and it is the time to attack her. Don't forget about Black Shield.
Morgana can only shield one champion with Black Shield. When she casts it on herself or her ally, lock the other one for probably effective gank.
Thanks to Black Shield, Morgana effectively counters champions like Thresh or Leona. Make sure not to take these champions when facing her.
When out of Dark Binding, Morgana is suspectible to poke, thanks to her low attack range and effectiveness of Tormented Soil versus unrooted targets. When as support, Sona, Soraka and Brand are somewhat effective versus them. Note that she still can absorb any disables with her Black Shield.