I for one, if i had to pick a single champ and call it the worst, I would say Sivir. I have never seen a good Sivir, nor have I seen a Sivir go positive. She really only has a single offensive abillity, an unimpressive farm tool, and her shield is pretty weak as far as I've seen. And not to mention her ult is just another steroid, how boring. I don't doubt that she could be good, I just think that anyone else can do her job better.
Sivir played properly is a wrecking machine.
Single offensive ability? Let's analyze this:
Q does a large chunk of damage on travel and return..
W resets her attack animation and allows her to auto attack 2-3 times in quick succession.
E is her shield, which can be used to bait and negate enemy CC/abilities.
R is an attack speed and movement speed steroid for her, her entire team, AND any friendly minions!
Unimpressive farm tool? Have you seen how quickly Sivir can push, and then demolish towers? Apparently not. Q and W combined with (maybe) a few auto attacks completely clear a wave, and W combined with R not only allow you to make short work of a tower (esp with friendly minions in tow), but her R and E combined with Flash/Teleport or Flash/Ghost give you a reliable means of escape.
Weak shield? Again, clearly you've never played versus a GOOD Sivir. Try laning against a decent Sivir as Taric, Alistar, Caitlyn, Teemo, Nidalee, Ashe, etc. As any of those champions, you use abilities which you either place on the ground, or which are predictable to shield.. Every time you try to use these abilities on Sivir or in a lane with Sivir, you run the risk that you just wasted mana, and gave Sivir mana, and then opened yourself up for counter-attack because one of your abilities is now on Cooldown and she has mana to spare. I've actually initiated fights for my team as Sivir by intentionally baiting an enemy tank to CC me, eating their CC with the shield, and then using my Ulti, which signals my team to initiate while I auto-attack myself back to the back-lines OR chase down runners.
Her ulti is NOT just another steroid. Her Ulti turns her entire team and any minions around them into a base-wrecking ball.. It also, as mentioned above, and be used to initiate team fights or help your team escape them when they're not favorable.
If anything, I'd say Sivir and Ashe probably compete for the top 2 spots for "most useful AD Carry to have on your team if played properly".