Recommended Solo Lane Choices (assuming you are laned against 2 opponents):
Alistar, Anivia, Blitzcrank, Caitlyn, Cho'Gath, Dr. Mundo, Galio, Gragas, Irelia, Lee Sin, Malphite, Malzahar, Maokai, Mordekaiser, Morgana, Orianna, Shen, Swain, Urgot, Vladamir, Yorick.
Most of the champions listed above make the list because they are tanky and have some survivability. Survivability can include: self heals, cc, natural regen, high hp growth, good base stats and the like. Caitlyn, Malzahar, Morgana and Orianna may not fit that "tanky" mold but they excel against two enemies for the following reasons.
Caitlyn has a long natural range, built in harass with her innate abilty, traps to control the lane (making two enemies slightly less of a problem) and Piltover Peacemaker offers great harass and farming. Another important feature to the solo laner is the rate at which they level up to 6 and get their ult (relative to their opponents). Caitlyn's ult is quite good and makes her a prime choice for any solo lane. Opponents to watch for include Teemo (he can blind you and his poison is no fun when you are on the receiving end) and the other tanky champions with good harass listed above.
Malzahar has lots of aoe damage and replenishes his mana with his Malefic Visions ability (provided it jumps 4-5 times). I tend to build Rod of Ages and Rylai's Scepter on Malzahar (sometimes even a banshees) each of which gives him some hitpoints so he isnt too squishy. Malzahar also has a great ult that can tear down an enemy champion if their lanemate doesn't disable you in time. Watch out for enemies like Soraka, Taric, Sion, Cho'Gath and other with silences or hard stuns as you will find yourself disabled too much to make a difference in your lane.
Morgana and Orianna both have excellent, long ranged, aoe abilities and shields. This combination actually makes them very tough to take down and allows them to farm and harass without much of a problem.
Avoid Soloing as:
Any support champion as you should be supporting someone which you can do if you are in a lane by yourself. Another sub category are the stealth/assassin characters. Evelyn and Twitch are more annoying in a lane with a partner because your partner can keep up the illusion that you are there by applying pressure to your lane while you go off and gank elsewhere. If you are in a solo lane you risk spending too much time invisible and not enough time farming and harassing. Also, you can expect to be jumped on when you unstealth instead of making them panic because they know they have you outnumbered.