I have recently seen a ton of people rushing Sheen on Vayne, and I just wanted to elaborate on how bad it is (compared to the alternative) by no means am I bashing Trinity Force, which is a solid item for her - Just sheen. It was presented in the spotlight by phreak, but I really think it's a silly item for her.

1) Tumble has a 1.75 AD ratio at level 5. It deals your total AD, plus 75% extra (100%+75%=175%)
2) For just 340 more gold you could get a BF sword that would not only add a superior amount of damage to your tumble(79AD), but also to your autoattacks(45) and your Condemn (a bit unsure of the Ratio, I think it's like 0.25 or 0.5 if impaled)
3) Vayne has like 65 base dmg at lvl 5-6ish which is the earliest without firstblood that you would probably get sheen. Would you rather add 65 dmg to your tumble for 1260 gold, or 79 dmg to your tumble, 45 to your autoattacks AND 22 to your successful condemn for 1600 gold?
4) Sheen AP is completely wasted
5) Vayne is not that mana hungry

So yeah. Get a triforce, sure, but don't get sheen just for it's damage component. Start with Zeal and Phage first instead. If you want early game burst damage, you should consider either a BF sword or even just a brutalizer.

That's assuming you can have rank 5 Tumble at "lvl 5-6ish", which isn't possible.

Regardless, yes, a BF Sword first will do more damage then a Sheen can do, but that isn't the reason for getting Sheen.

Sheen first is a cheap damage boost to your tumble, which lets you jump forward, poke, then get back taking very little or no damage in return if done right.

Sheen first lets you quickly get to another important "Trinity Vayne" item: Phage. Phage gives some much needed survivability and the ability to start kiting effectively. It doesn't matter how sweet your BF Sword auto attacks are if you are getting hammered with damage in return. The slow on Phage, plus the health bonus might seem insignificant, but they are very helpful in the beginning phases of the game.

Sheen first lets you build into a faster Trinity Force. The move speed from Zeal, the burst proc of Sheen, and the survivability of Phage outweigh the benefit of overall increased damage from BF Sword. Then, when turned into Trinity Force, you get all those benefits, plus amazing damage.

I think the OP has done his homework.
/golfclap
+1

Trinity Force is the most expensive item in the game, and for the amount of money you sunk into building that item, there are much better uses.

If you are farming fast enough for a TF, then you might as well get your boots and an IE and really do damage.

Spammable roll and AD-AS dependent?

Why yes, I would like a Manamune with that Black Cleaver and a side of fries instead of the Sheen.

Assuming level 5 on Vayne:
Sheen + Long Sword = 66+10+.65*10 = +82.5 damage on Tumble (Rank 3, max rank possible at level 5)
B.F. Sword = 45*1.65 = +74.25 damage on Tumble (Rank 3)
Sheen + Long Sword = 1260+415 = 1675 gold, builds into Trinity Force
B.F. Sword = 1650 gold, builds into Infinity Edge (probably)

And Trinity Force > IE for Vayne. 50 less damage, 10% lower crit, normal crits in exchange for +30% AS, +12% move speed, 250 health/mana, an on-proc slow and +150% base AD every few seconds? For only ~200 gold more? I'll take that any day.

Edit: Rushing Zeal instead of Sheen can be a good idea on Vayne if you're in a lane where you can consistently land consecutive AA's. Just like leveling Silver Bolts instead of Tumble can be a good idea in that case.
In most other cases where you can only poke, leveling Sheen for the proc on Tumble gives you more damage on harass.
Rushing Phage first is only done if you want the health.