Use the info here with consideration. It might be outdated or biased info.
Lanes and Roles
Morgana is considered to be strong in two roles; as an AP caster at mid lane, or as a disengage and poke-based support.
The latter role is, at the moment, the most common, for a variety of reasons; first, Morgana's utility does not rely on her building AP. Second, Morgana is a counter to various popular CC-based aggressive supports, such as Thresh and Leona, due to her Black Shield and her harass with Tormented Soil. Dark Binding is also an extremely good CC with long duration, and while somewhat difficult to land, it is very powerful against the enemy ADC and is detrimental if it hits, as it usually results in a kill if the enemy does not play carefully.
However, she is also viable at mid lane as a mage. There, Morgana takes advantage of her damage potential with Tormented Soil, Dark Binding, and Soul Shackles, as well as her ability to have an extremely safe laning phase due to her spellvamp and the fact that Morgana is difficult to gank due to her Black Shield. Morgana is a good counter to some mid lane mages, although she must watch out for AD assassins, which are her best counters, even though she has far more teamfighting presence and utility than them.
When going mid lane, Morgana can afford to rush Zhonya's Hourglass as her first item. This is good as when going support, she usually has to buy other items first, and can't rush an early Zhonya's Hourglass, which means that if Morgana uses her ult and is focused, she risks dying before she is able to stun her enemies.
Use Dark Binding to check for enemies in the brush. The projectile stops moving and makes a distinctive sound when it hits something. While this method does work, it is best to save Dark Binding unless you have maxed cooldown reduction, as it is an essential ability for Morgana's combat and will leave you vulnerable if it is on cooldown.
Likewise, Soul Shackles can be used to detect stealthed units and opponents in the bushes. The icon of Soul Shackles will become available only if an enemy champion is near you. It can be used to see where they are the moment you use it, but you can't keep track of them if they are moving.
With enough ability power, Dark Binding can deal a fair amount of damage as well as snare the opponent. The projectile moves fairly slowly, though, so always lead your target at any distance longer than close range.
Dark Binding is an exceptional zoning tool. If you're being chased, throw it at your enemies so you can get away. Even if you miss, it will cause enemy players to waste time moving around it.
Tormented Soil, used on an enemy champion and following up immediately with Dark Binding to snare them in place, is an excellent harassing tool in the early game. Low health champions can even be killed outright since Tormented Soil deals more damage to lower health champions and will hit for its maximum damage for every tick rather than grow in damage when the target is at much higher health.
Note that Tormented Soil deals a tick of damage immediately when placed on the ground. Use this knowledge to finish off an enemy fleeing with very low health.
When using Tormented Soil to harass an enemy, place it somewhat behind them. This is because they will naturally run backwards, and doing so will cause them to take at least 2-3 ticks of damage. Eventually, they will catch on and run to the side or forward; use this to your advantage when casting Dark Binding.
If you level it up early, by rank 3 you may be able to clear a wave of caster minions with Tormented Soil, and possibly even melee minions, depending on ability power and time into the game. As the game progresses, minions receive additional health and magic resistance, so either level up the ability, or build more AP and/or magic penetration.
Each tick of Tormented Soil, on every single enemy champion hit by it, procs the Tribute passive from Spellthief's Edge. This is a quick way to maximize gold generation.
If an opposing champion must be killed in a team fight, use Soul Shackles and Dark Binding on that champion - if it doesn't kill them outright, it will at least deal a significant amount of damage and leave them stunned.
Flashing into a group of enemies, using Soul Shackles and then Zhonya's Hourglass, is excellent to initiate team fights, stun many enemies, and come out with a fighting chance since they will likely be trying to scatter away.
Black Shield is a vital ability to learn as it stops all forms of crowd control and also makes your target temporarily absorb magic damage. Shrewd use of Black Shield can determine the outcome of team fights.
Try to cast it pre-emptively on your teammates when they initiate, or as soon as you see them targeted by any spell or ability. If you can get the shield in before your teammate is hit, it just will absorb it.
It will still block the crowd control effects from Physical Spells and on-hit items, however.
If you play Morgana more offensively than supportively (at mid, for instance), it is best to go for higher levels of her other spells for farming unless you are against a heavily Crowd Control orientated team, as you will need damage to lane effectively.
Morgana is a magic damage caster with strong AoE damage and disables and scales well from ability power.
Zhonya's Hourglass is a core item for Morgana; in addition to giving armor and massive ability power, the active is absolutely invaluable as it can be used while Soul Shackles is still casting and with proper positioning she can hit multiple enemies that would otherwise kill her while staying alive to finish them off after it’s over.
Rabadon's Deathcap is great on most every mage champion. However, Morgana is best if built a little tougher first and thus it is recommended to delay purchasing this item until at least one other item is finished, though it is not needed directly.
For enemies building high magic resist, building a Void Staff can help negate their MR and provide more damage.
The magic resistance from Abyssal Scepter and the added ability power and magic reduction aura make this item a decent anti-magic damage item but just remember her two primary damage spells far out range the magic reduction aura it provides making it only useful at close range (team fights and melee attackers).
Morgana is required to be in the center of a fight so that she can hit Soul Shackles on everyone. Thus, items such as Frozen Heart or Randuin's Omen are good items if you are unable to survive long enough to be effective in fights.
Rod of Ages is a very good item due to the good amount of health and mana it provides, albeit it takes some time to fully charge up so it is best when purchased as a first major item.
An early Catalyst the Protector provides some good early sustain during the laning phase as well as the mana and health keep her in the lane for extremely long periods of time making this item perfect for early, mid and late game.
Morgana tends to quickly barrage her spells even if Dark Binding fails to land; without it you will be able to avoid her other abilities and kill her.
Avoid standing above Tormented Soil's AoE. The spell damage is small but the tick is very fast, making it a dangerous spell.
This spell can no longer reduce your magic resistance. However, it is now a lot more dangerous if you happen to be at low health.
However if you are a very fragile champion moving out can easily take you short time of health, bursting her down or outmatching her regeneration may be wiser.
Morgana's high survivability makes her dangerous to DoT based mages such as Swain, Cassiopeia or Anivia as she will be able to output the damage long enough to kill her opponent, but is weaker against potent disablers with high burst damage as Malzahar or LeBlanc who can ignore her spell vamp and have an easier time breaking her shield.
Remember that upon activating Soul Shackles no CC will be able to stop the second hit, silences are worthless in such case, but all other forms of CC will cripple her ability to keep you in range, however she can eliminate this problem with her shield. In other words, it is better to use some form of dash or speed up to escape the tether.
When Morgana uses Soul Shackles, quickly move away to break the spell tether and avoid the second damage and the stun.
This can be hard however as the spell comes with a slow, remember that killing her will also break the tether.
Morgana's most dangerous form of attack is opening up with Dark Binding and immediately following up with Tormented Soil and Soul Shackles—at most stages of the game, this combination can be devastating and in most cases it spells instant death for the one being focused. Taking great care to dodge her Dark Binding will greatly reduce her danger to your team, as that very same spell is also her method of escape.
Due to Morgana's anti-caster nature, consider using a physical damage champion, such as Kha'Zix or Zed, against her.
However as any burst mage she can kill a fragile champion outright so sending a fighter is a more reliable choice, making ADfighters a reliable choice even in lane.
The aforementioned Kha'Zix and Zed, however, are two AD assassins who can outright kill Morgana and they have reliable ways of escaping of her Soul Shackles (using Evolved Wings and Living Shadow), respectively. They are also very mobile (it is difficult to land Dark Binding on them) and because they are AD and have no CC, Black Shield is useless against them. This makes both Kha'Zix and Zed two of the best counters to Morgana, with Kha'Zix having an advantage because he has some sustain with Void Spike so he can try to outsustain Morgana's spellvamp.
Zhonya's Hourglass is her only means of survivability during teamfights. Once she activates this item (or if for some reason she didn't yet bought this item), Morgana is easy to focus and kill. Remember that she is very squishy.
Even if Morgana is going support, don't understimate her damage. She has massive base damage on her spells.
If Morgana is going support, avoid picking tanky melee supports with a lot of CC such as Leona, Thresh, Blitzcrank, or Braum. This is because her Black Shield will block their CC and they either don't have healing or their healing is too weak to outsustain Morgana's ranged harass.
If the enemy picks Morgana into your tanky support, however, there are some tricks you can do to at least survive against Morgana. For example, try to bait Morgana into using her Black Shield, then focus either herself or her ADC (focus the one who is NOT affected by Black Shield).
However, Morgana is weak against poke based supports such as Lulu and Karma, because she can't protect against their poke. Morgana herself does not have any means of healing, and while her Black Shield can protect against their poke, it has a long cooldown. Poke based supports also tend to have means of disengage, to counter against Morgana's Dark Binding and her engage with Soul Shackles.
She is also weak against supports with healing, such as Soraka and Sona, as Black Shield has little use against them, and her harass can be healed.
Take advantage of the fact that support Morgana can't afford to rush a Zhonya's Hourglass (because she needs to buy other items that are core for supports, such as Sightstone) and focus her if she activates her Soul Shackles.