What situations/champions are Manamune designed for?

Manamune is an item usually for AD Casters who use single target damage/weave auto attacks between their skills.

The problem with this item is that it staggers your laning phase with most melee champions because you have to buy the tear first and then the item, which in turn doesn't give any good mid game stats.

Jayce gets it is to relieve his mana-costs, easy-stacking, his W in ranged form. Manamune is also good for single target damage like Kha'Zix, but it's difficult to be a jungler and get that item as well without sacrificing your mid game power.

So the champions you want to build this on: Blue Ezreal, Jayce and Urgot because they weave autoattacks and abilities almost constantly. It's funny but Ryze also enjoys the manamune because all of his abilities are single target and scale off of his mana. But why go that route when you can go Archangel's staff, right?

Most melee mana-users have better mana sustain than ranged casters, and are better to rush other items like Brutalizer or other melee items. And since most AD Melees are also considered fighters, they get things like Frozen heart or Iceborn Gauntlet and that relieves the mana issues too.

I noticed that nobody mentioned Yorick yet. He is the perfect example for a melee AD caster to use Manamune, he can almost indefinitely sustain himself with his E, making him quite hard to beat in lane. Spamming his skills while also landing auto-attacks with his Q stacks his tear quite fast.

Manamune/Archangel's Staff is mostly designed for champions that benefit from large amount of maximum Mana or are very manahungry (AD casters).

Champions that scale have abilities that scale with mana:

• Ryze: All his abilities, but the ultimate, scale with max Mana.
• Blitzcrank: He's very manahungry and his passive scales with the amount of Mana he has.
• Kassadin: His ultimate (Rift Walk) scales with max Mana.
• Singed: Singed gains bonus health equal to 25% of his maximum mana.
• Evelynn: While stealthed, Evelynn regenerates 1% of her maximum mana every second. (there are definitely more viable items for her)

Other:

• Karthus: He's really really manahungry.
• Ezreal/Urgot/Jayce/Yorick: They deal mostly more damage with his abilities rather than auto-attacking.
• Pantheon: Is also a melee AD caster but there are definitely more viable items for him.

Since you are asking about melee champions that will use manamune effectively, I can only think of two viable champions:

Yorick: Top lane, you need to get tear of the goddess ASAP it is very important for him to have it for harassment in lane with his ghouls, it gives you great sustain to be able to spam Yoricks E, which gives him more hp on use and when the ghoul itself attacks anything.

Kha'Zix: In top or mid lane, never jungle. Almost for the same reason Yorick needs it, allows you to have great sustain and harassment in lane be allowing him to spam his W which heals him when it hits a minion or champion and you are close enough for the effect to heal you.

With any other melee champion there are better items to get then manamune in lane or in jungle. This is all based on gold efficiency for particular champions.

While many of the answers here have listed very good champions for the manamune, I find that many people underestimate the power it can have with Fiora.

"But Warped, taking the tear ruins Fiora's early game lane dominance!"

Yes. Yes it does. And in some cases, you don't want to be so aggressive early on. Mundo, for example, is a very good Fiora counter in my experience, because it is very difficult to burst him down and his AOE will destroy you while you're near him. With champions like Gnar, where it's difficult to get close and stay close, a more passive approach also works.

So, with all that being said, on to the mid-late game. The only unusual items in this build are the Manamune and the Essence Reaver. Aside from that, a completely normal Fiora build works fine, Lucidity boots, Cleaver, Last Whisper, and Ravenous Hydra. All together, that gives you 35% CDR, enough to use abilities frequently, armor pen, and lifesteal, as well as mana regen on hit so you can keep the Manamune on all the time.

This build can be extremely strong in the right circumstances, unlike other champions like Yorick and Kha, where it is an extremely useful item no matter what's going on, and it certainly isn't essential like it is on Jayce and Ezreal. Although, I think it's important to say that taking Manamune on Ryze is pointless because the Archangel's and Manamune both use the same mana pool, because it is a unique passive on each item.

I'll be editing in numbers tomorrow, I hope this gives some thought for now.