Champ reworked, spell removed, masteries redone. Nothing about this build applies anymore.

Want to win games while having a deceptively mediocre k/d/a? Then look no further, Sivir is the best pusher in the game. This is a bit of a niche build, in that Sivir is wicked awesome at pushing but otherwise needs buffs or a rework. But for now at least, you can take that advantage and run with it.

Basically, you push open lanes whenever possible, teleport around as needed, and try not to die. Final item doesn't necessarily have to be a Bloodthirster, but if you want to really sucker punch their turret, that's the quickest way to do it. Alternately, you could get Warmog's or something for survival.

This build lets you be a hidden pubstomper. Higher-level viability needs testing, but this style of Sivir play absolutely cannot be safely ignored in any case. I pretty much went 6/9/14 my first game with this build and more or less single-handedly brought down two inhibitors and three turrets over the course of the game, and I finished the build in about 35 minutes. I had a creep score of about 300, with the game ending at about 40 minutes when I grabbed four elixirs and popped both of my team move speed steroids, after which our team just swallowed up that pesky Twitch in no time flat.

Philosopher's Stone Even if you're not l33t pro at getting free mana out of your Spell Shield, you can spam Q a fair amount all game long. Regrowth Pendant and a potion of your choice will help.

Might as well use that passive and the mastery tree bonuses.

This is a nice one. Debuff doesn't work on turrets (I don't think), but it synergizes like crazy with Zeke's Harbinger in 1v1 and in teamfights.

This is kinda nice. A bit underrated, this puppy will save your *** while granting you some supplemental AD. Because it's a cheap item, you want a short game. But you're the best pusher in the game, so it's cool.

This has an aura which is basically the only synergy for Ricochet in the game (other than AD of course). You can't apply on hit affects to an entire minion wave, but you can apply this aura.

Upgrade to this whenever you want. With all the farming you'll be doing, the GP10 won't be much missed. Pop this along with your ult in a teamfight and bam, your team can chase anything. Can use it to escape ganks after you've used your ult to take down a turret or two, as well.

Bloodthirster is great for maximum solo push destruction power. As a vanilla item for players who still want to hit hard, this is a decent choice. Solo gankers can't really expect a free kill, especially if they're only half-done with their build at this point.

Core item substitutes:

Instead of Ninja Tabi and Hexdrinker.

Pros:

Free wards can help you avoid getting ganked
You can grab buffs and solo Dragon that much more easily
Madred's Razor is strong early

Cons:

Lose out on dodge synergy
Magic damage poses a bigger threat
Difference in pushing power tapers off to nothing before too long

Since you'll be running away a *lot*, it's probably better to have Dodge trigger your move bonus as early as possible. With your ult and Shurelya's Battlesong, nothing but a stun or snare can stop you, and Spell Shield could save you anyway.

If you really want to play support style, this is a nice late-game swap-out for Hexdrinker. You lose a little AD, but you stand a much better chance of shrugging off ganks.

Last item substitutes:

Guardian Angel won't really make an impact unless your allies make excellent use of those few seconds, either in the form of saving you or pushing advantages elsewhere.

Warmog's Armor is a better item for safer teamfighting.

Teleport is mandatory. Revive or even Cleanse could possibly help out ( Shurelya's Battlesong + On The Hunt + Cleanse is a LOT of escape potential), or if you're on your toes, you can Flash out of ganks before they CC you. If you're pushing lanes, you WILL get gank attempts. Exhaust is nice if you want to hit with Q both ways.

I go with Rally here because ulting and dropping Rally with a full wave of minions on your side can really hurt a turret even if you have to run.

Pros:

Best pusher in the game
Best farming in the game
Forces them to choose between losing turrets and abandoning fights elsewhere
Somewhat sturdy
Nice supplemental damage in teamfights
Finishes build early, and then she can spam elixirs at whim.
Enemy stealth characters go squish extra fast in late-game teamfights when you pop your ult and Shurelya's Battlesong and you Spell Shield away attempted stuns.

Cons:

Countered to some extent by good map awareness, teleports, well-coordinated ganks and such. Twisted Fate is a good counter.
Has to push for a quick win or she falls behind as everyone else finishes their builds.
Character needs buffs to have much 1v1 viability in general.