There are two questions built into this: Which lane is the most carry-able (that is, easiest to carry) and which lane is the most important to have carry.
First, let's just come out and say that, given a sufficiently fed or farmed character, any role can carry. Getting into that position is more or less difficult depending on more factors than we can reasonably cover. However, I won't address those long-tail style situations and focus on the core of the problem. I'll also be assuming that everyone is of equal skill and the champions are all equally balanced for their roles.
Second, strong play is always going to be the defining quality behind "carrying." It just usually starts with getting a significant lead in the laning phase. This isn't a requirement, just a general observation.
Third, a quick TL;DR:
- 1v1 lanes are the easiest to carry because there are fewer factors that can result in lane dominance.
- If your goal is to ace and then push, it's more important your AP role is carrying. If your goal is turret pushes or siege-based strategies, then you'll want your AD carrying to maximize turret damage.
In which lane is it easiest to carry?
1v1 and 2v1 lanes are definitely the answer here. The reason being that in a duo-lane you have someone else looking out for you. Either they're a support and they mitigate anything that happens to you (either by enabling kills of your own, mitigating damage, or healing) or they're another power-house which enables you to go back, heal, and buy items. However, in a 1v1 lane, if you can establish dominance, your opponent has few choices but to play cautiously and wait for you to make a mistake. Jungle ganks are the equalizer, here, but given equivalent quality and quantity of jungle ganks, the person in the lead is the one who will dominate that lane. It also means that you're getting more gold and CS than your opponents and allies. The faster you're getting your levels and items, the more you're going to carry.
This means that, traditionally, both top lane and mid lane are going to be the easiest to carry in, all else being equal.
In which role is it the most important to carry?
I swapped to "role" rather than lane just for clarity, no other reason.
This question hinges on how you intend to destroy the nexus. Do you siege turrets and inhibitors, disengaging whenever the enemies approach? Are you going to ace the opponents and take the opportunity to take a turret or two? Are you just going to bully through with multiple tanks and push turrets like dominoes?
All of these are perfectly valid strategies that fit within the current meta. The most common by far is acing the other team and then pushing, of course. There are too many variables to effectively cover the math in this situation, but in general you'll find that AD carries will do more damage to turrets than AP carries will. Why? It's all about damage per second to a turret.
Source
Damage to a turret is based on your base attack damage plus the higher
of your bonus attack damage or 40% of your ability power.
Let's take a base case that you're playing Annie who does (49 + (2.625*18)) * 0.712 = 68.53 damage per second at level 18 (ignore runes, masteries, and that "AD Annie who really carried that one game"). Let's say you have 1000 AP, which we can all agree is exceptional (both in the degree of its quality and the fact it's an exception to the norm). That means you'll be doing 68.53 + (400 * 0.712) = 353.33 raw damage per second to that turret that has 50 armor (for the outer turret; I don't have data for others).
It doesn't take much to imagine an AD carry being able to dish out significantly more raw damage per second to that turret, even taking into account that critical strikes and armor penetration don't apply. Let's all just pause right here and also recognize that adding a Lich Bane to that AP build would tip the scales significantly. But that only covers some AP carries, and I'm trying to be general.
Clearly, if your goal is to push down turrets, either by using a bully-tank based approach or a siege, you want your AD bot to be carrying.
If you're looking to kill the enemy team, however, an argument can be made that having extreme burst damage that you traditionally get from an AP carry makes the most sense. This math is less straight-forward as the turret's static values aren't in place. Suffice to say that the burst from your AP role in the mid-game is designed to be more powerful than you AD, and is often area-of-effect, increasing the total damage done to ridiculous proportions.
In short: Late game and siege-based teams would do well to ensure their AD is well farmed and prepared to carry, whereas roaming gank squads and kill-based teams should focus on making sure your AP is ready for action as soon as possible.
Best Answer
Getting CS is about much more than just last hitting. In fact, solo laning itself revolves almost strictly around last hitting in high level play. You need to know how to balance all of the following:
Last Hitting: the skill of knowing how much damage you do to a minion to ensure that you obtain the gold from killing it.
Keeping the lane where you want it: This is a huge factor, some champions like Malzahar and Morganna can push harder than champions like Ryze because of their AoE abilities so should the Ryze player wish to keep the lane in the middle he must attack the wave more than just when the last hit is available.
Some champions however excel at killing minions under their own tower and may want to keep the lane there for safety while others have many escapes and can push to the enemy tower all they want and still escape back to their own should the enemy jungler gank. Part of mastering your favorite champions is learning how to balance risk vs reward with your available tool set.
Harassing: You also might not want to only keep an eye on the health bars of the enemy minions, you should also keep an eye on your own. This is especially important in top lane where the enemy usually last hits in melee range. Therefore you know where the enemy will need to be in order to get that gold. If you move or attack in such a way that they cannot reach that low hp minion without taking significant damage they will miss the gold or better. The #1 rule of hunting is: attack your prey when it attacks it's own.
Counters: Most champions have a wide array of other champions that they do particularly well against. Counters usually are formed when a champion excels in one of the above areas that their opponent lacks.