Not Updated For Current Season
Heads up: This is my first guide, as you can see if you've got some brains. You could see it yourself? Good, you might have the brains needed to play Sivir.
Some words first, my own thoughts about Sivir, is that she's, if played right, not the carry of a team, but the entire backbone. A carry is someone that deals with kills only, if you ask me.
Playing as Sivir, you're never strong enough to be 'just a carry', because you're lacking offensive abilities. Sure, you've got the Boomerang Blade, but that'll never be enough to compare to others that might have 2-3 abilities that are offensive.
What you do as Sivir, is you keep annoying your enemies 'til they die, or you get help from your teammates. Save Summoner spells for a tight situation where it benefits the whole team, because, let's face it, you can never be the one that kills a whole team that's on the same level and got equal items to what you've got.
So, with that said, let's get down to business, starting with Pro's and Cons.
Possibilities for heavy damage
Many chances for high movespeed
Spell Shield can be a con if not used right
Boomerang Blade easily avoided
No escape mechanism
Play Style: To me, Sivir should always be in every single teamfight, possibly in the front as well, soaking up possible roots from Lux, Morgana etc. using the spellshield, making sure that the team can keep being mobile at all times. To me, the summoner spells Exhaust and Rally are tools to keep Sivir being an even better teamfighter, considering Improved Rally also grants ability power for your casters as well.
When it comes to items and damage, I have before preferred Attackspeed-builds, but as of late I've realised that when it comes to Sivir, you go over the attackspeed cap (2.5) using the ulti, which literally makes the ultimate useless, except for some increased movespeed.
After I stepped away from that, and after they changed the Boomerang Blade to increase from attack damage, another build with Sivir came avaliable; the massive damage build.
However, as I've said earlier, Sivir is fragile, thus your question comes to "Why don't you build any extra survivability?". Because building her extra bulky screws up the damage she does, and considering she only has Boomerang Blade as a damage ability, she looses A LOT of damage, and almost is rendered useless to the team, except for the ultimate, and let's face it, that ain't that great, unless you're a full team of carries, and that doesn't happen often.
Why these spells? Because, as I've said, Sivir is ment to be a team champion, and thus, I play her like that. Ghost, to either slow down someone for a team kill, or render a enemy Carry useless vs a team member or myself. Improved Rally grants both attack damage, and extra ability power for the casters of the team, probably giving you a upper hand through teamfights, unless the Rally-standard gets sniped down (mostly that never happens, even though it gives a ridiculous amount of extra damage to the team).
Going with the 21/9/0 build, because, as I've said, Sivir is a AD-teamfighter, so I pick everything in the offence-tree that improves physical damage, along with Improved Rally and Improved Exhaust.
In the defence-tree, I go down and pick the Nimbleness-talent, because of the increased dodge Sivir gets while moving, it provides a slight possibility to escape, but first and foremost, it grants you a way to towerdive. If you attack a enemy champion, you will be moving, and you will have minions attacking you. If those attacks are dodged, you will get another 10% movespeed for 5 seconds. This gives you a chance to catch up with enemy champions using ghost, when you yourself are using Rally, and also benefit from the Nimbleness. You can also towerdive easier, because you will be able to run in, get the kill, and run out, hopefully fast enough to not get slapped in the face by the tower too much.