Tabulating the exact number of fights in either Avatar: The Last Airbender or The Legend of Korra is an exercise in futility — there are simply too many characters in too many situations that end up in some kind of conflict or another. Further, since the overarching themes of both shows revolves around friendship, many battles are fought as a group rather than individually.
However, taking into account only individual victories (and excluding those won with the help of others), it might be possible to estimate which member of Team Avatar has gotten into the most scrapes. Interestingly, the number remains more or less balanced across either series.
10 Suki - 2
The gang first comes across Suki on Kyoshi Island, where they are captured by the Kyoshi warriors for trespassing. Considering that Suki is the head of the group, the victory for this match goes to her. Also, she later helps them escape from the invading Prince Zuko, so that should count for something.
Finally, during Sozin's Comet, Suki arrives in the nick of time on one of the last remaining Fire Nation blimps, thereby saving Sokka and Toph from a rather painful death.
9 Asami - 2
Asami might not be a bender, but she sure fights as well as one (and often better, to be honest.) She initially refuses to consider the possibility that her father, Hiroshi Sato, might be secretly entangled with Amon's terrorist group, but when the fact is made clear, she shows no hesitation whatsoever.
Asami stuns her father with a bolt of electric current and helps her friends escape. Later, she travels with Korra on a mission from Hou-Ting of Ba Sing Se, to collect taxes on the Earth Queen's behalf. Here Asami encounters a group of "bandits", many of whom are dispatched by a combination of her electrified glove and acrobatic skills.
8 Mako - 3
Mako's first solo fight occurs against Amon, right after the villain extracts the three elements out of Korra. Before the same thing happens to him, he forces his body to shoot a lightning blast at the Bloodbender, temporarily knocking him out.
One of his best accomplishments, however, is against the prodigious Waterbender, Ming-Hua, against whom he has to rely more on tactics and wit than pure might. Finally, Mako single-handedly destroys the core of Kuvira's spirit vine-empowered platinum mech, causing it to short circuit.
7 Bolin - 3
During the premiere of the concluding segment of Nuktuk: Hero of the South, a few men enter the President's cabin to kidnap Raiko, but are thankfully foiled by Bolin's quick-thinking. Against the Red Lotus, the young Earthbender finds himself horribly outclassed, but this only lasts until his random and epic discovery of Lavabending (that he then uses against Ghazan at the latter's hideout.)
Most importantly, he uses specialized Lavabending techniques, such as his flaming disc, to defeat the soldiers guarding the innards of Kuvira's mech, giving way for his brother to perform his own task.
6 Sokka - 4
The first of Sokka's wins isn't technically a battle, but the fact remains that it is he who smartly turns the pirates against Zuko and his men, claiming that they would get a lot more reward for the Avatar than for the Waterbending Scroll they so desperately want to retrieve. In the Northern Air Temple, he and the Mechanist construct the first air balloon in history, later using its flight capabilities to accurately "bomb" the attacking troops.
Obviously the most impressive feat he has ever carried out is making Combustion Man annihilate himself, to the joy of both Team Avatar and the audience. At last, one cannot forget his brilliant assistance in dismantling the Fire Nation airships going to invade the Earth Kingdom.
5 Toph - 4
Toph is literally introduced as a winner, specifically of the Earth Rumble in which she effortlessly defeats The Boulder (to Sokka's utter astonishment.) When Xin Fu gathers up his men and kidnaps her and Aang, she goes on to fight all of them simultaneously, making her Earthbending teacher, Yu, exclaim that she was "the greatest Earthbender" he had encountered.
Toph garners a small victory against Combustion Man in Fire Fountain City, when she accurately smacks his forehead eye tattoo with a sharp pebble and causes his own attack to backfire. Also, she practically rips enemy airships to shreds during Sozin's Comet, prompting Sokka to declare his happiness at her inventing Metalbending.
4 Zuko - 5
In the storyline, Zuko's primary win is against Commander Zhao, in an Agni Kai, although the loser is shameless enough to attack the prince when his back his turned. He goes on to "rescue" Aang from Zhao and his several soldiers, including a band of Yu Yan archers, while disguised as the Blue Spirit (with Aang having to save his life at the end.)
Zuko attacks Katara at the Spirit Oasis while Aang is in his inert state; he succeeds in overcoming her Waterbending after much difficulty. At one point, he abandons his uncle to go roaming the Earth Kingdom by himself, where he gets into a brawl with a few deserters, whom he teaches a well-deserved lesson. His concluding battle is against his own father, but his only technique, Lightning redirection, happens to be a defensive one.
3 Katara - 6
Katara pretends to be an Earthbender to get herself thrown in prison along with Haru, his father and other Earthbenders, where she exhorts everyone to fight back. Subsequently, she defeats Jet in a single move after Sokka reveals that the rebel had some pretty horrible intentions in mind. Further, when Appa "crashes" into the Foggy Swamp, Aang and Sokka and more or less helpless before the Swamp Creature, but Katara literally unmasks it (they do become friends, though.)
She insists on helping the poor villagers of Jang Hui as the Painted Lady, a job she almost completes solo before her friends join in. During their so-called tour of the Fire Nation, Team Avatar comes across Hama, who depicts herself as a kindly Waterbender but then suddenly shows her true colors, forcing her to take the old woman down. Finally, Katara defeats Azula with a series of rather ingenious moves, the likes of which have rarely been seen in the story before or since.
2 Korra - 8
Korra's first incursion into Republic City gets her in trouble with Lin Beifong because she pretty much destroys a group of criminals trying to extort money from shopkeepers. She displays her natural talent for Waterbending in the Pro-bending matches; in one, she takes on a whole team by herself when Mako and Bolin are knocked out of the ring. She routs the Equalists trying to re-kidnap her from Tarrlok, and with just her Airbending, exposes Amon for the complete fraud that he is.
When the Southern Water Tribe rebels foolishly attempt to abduct Unalaq, she blocks them from escaping and liberates him (although she does regret it later.) The climax of Book 2: Spirits is all about Korra's spiritual projection infusing UnaVaatu with the light of Raava, consequently killing her uncle. Along with Asami, she overwhelms the bandits endeavoring to steal a large amount of gold. She technically loses to Korra in Zaofu, but restores her honor by defeating her inside the platinum mech.
1 Aang - 9
Zuko makes Aang his prisoner in exchange for leaving the Southern Water Tribe alone, but the Avatar quickly throws off his captors and flees on Appa (right after instigating the Avatar State for some incredible Waterbending.) His victory against King Bumi is counted, because the latter un-Jennamites them as promised. He helps two opposing factions cross a dangerous canyon, where he figures out the best way to escape and protect themselves from the canyon crawlers at the same time. Aang combines his powers with the Ocean Spirit and washes the Fire Nation armada away with a single wave.
Also, the villagers of Chin sentence him to death for something Avatar Kyoshi did in the past, but he magnanimously protects them from the Rough Rhinos. Aang "participates" in the Earth Rumble, defeating Toph and winning the championship belt (but he cheats by using Airbending.) While taking a break from Earthbending practice, he Earthbends an enraged Moose Lion mom away from Sokka, obtaining Toph's praise for the first time. Furthermore, he is the prime mover against the Drill attack on Ba Sing Se — he concurrently stops the machine and Azula, a rather impressive accomplishment. And then there is Ozai.