Made to be as to-the-point as possible. Weapons listed in the order they appear at the workbench, abilities in no particular order.
If you believe something is incorrect or could be clarified, please note it in the comments.

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Found at the end of Act 1, Stage 2 (Depth 2), after the miniboss.

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Found at the end of Act 2, Stage 2 (Depth 6), after the miniboss.

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Found in the Pregame Lobby, just to the left of the workbench.

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Found at the end of Act 2, Stage 4 (Depth 8), in the room after defeating the boss.

The door to the weapon (red) opens after destroying 4 crystals in this room.
Crystal 1 is to the left of the door, past a grate (green).
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Crystal 2 is nearby in one of the upper grates.
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Crystal 3 is in the grate on the right side of the small pond.
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Crystal 4 is in the vertical grate near the exit tunnel.
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After destroying all four crystals, the door opens. The order they are destroyed does not matter.

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Found at the end of Act 3, Stage 2 (Depth 10) after the miniboss.

The mural on the right side will vanish if you've defeated the miniboss quickly.
Only time taken on the miniboss influences this, total run time is not factored. I estimate the time given is less than 1 minute.

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Found at the beginning of Act 4, Stage 1, before you drop down from the starting area.
Your total runtime must be under a certain amount, estimated to be under 40 minutes.

At this point in the level, turn around.
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Cross this gap (jump, dash, jump) and proceed. If you've made it in time you'll be able to proceed to the room containing the weapon.

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Found in the Pregame Lobby, in an out-of-the-way room.
The "weapon" will spawn after you've killed every miniboss variant in Act 4, Stage 1 (a random one out of three is selected each run).

To find the room, follow the seaside path found right behind spawn. The path leading to the room is behind the waterfall.
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You'll eventually come upon this room. The far wall will glow and allow you to interact with it to open it.
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After entering the room, wait a little until the statues activate and fill the pond with water, after which you can find the Fish Deity swimming circles in the fountain. Again, you must have killed every miniboss variant in Act 4, Stage 1 for the statues to activate.

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Found in Act 1, Stage 4 (Depth 4) after the boss.

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Found in Act 2. During a transition area between encounters, there may be a locked door. Opening it with a key will let you access the ability.

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Found in the Pregame Lobby, in the 'trophy room'.

The entrance to the trophy room is opposite the workbench, past the soul upgrade wheel.
The area pictured below will be accessible after getting 6 artifacts, which are acquired from completing challenges. If you've done a few runs or gotten past Act 2, you likely already have enough.

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Found in Act 3, in a transition or starting area.

The 'safe room' with multiple pillars surrounding a bridge will have the ability at the top of the tallest pillar. You're likely to come across it multiple times throughout Act 3.