Sorry, but I think that Katara would most likely lose against Ty Lee, unless she reaches a level of mastery high enough so that she can bend TY Lee's body fluids. The only way Katara could ever win was if she had an enormous amount of water nearby, and Ty Lee is still far away.

Now, a better match for Ty Lee would be Aang, because he's just as fast as she is. (Sorry that was off-topic, but I had to say it.)

Katara is quite powerful, no doubt. According to the Master at the North Pole, she's already achieved a mastery status in very little time. But Ty Lee is the most agile fighter...quite honestly that I've ever seen (in an anime/cartoon, obviously). Ty Lee has a large advantage, because not only is she agile, she's quick, and as powerful as Katara is, she doesn't have the same reflexes. We've never seen Ty Lee serious in a fight, as well.

Katara would potentially have a chance with the 'Octopus Style' as it completely surrounds her, making any approach very difficult. But in one of the last episodes, where she comes upon the three still dressed in the Kyoshi outfits, Ty Lee takes her down with a single tough, and fast too.

I personally think that Ty Lee has the potential to wreck any of the benders, but she's my favorite female in the show, so...*shrug* That's just my opinion, which isn't quite valid in an argument like this,

Part of the problem with the Kyoshi infiltration scenario is that Katara didn't actually stand her ground and fight. She turned and tried to run, and when you're running away from someone faster than you, well... It doesn't go over too well. If she hadn't turned her back on Ty Lee, she might've lasted a little longer. Maybe. I don't blame her, though. She was quite outnumbered and quite outmatched with all three of the "Omashu girls" as her potential opponents. The smart thing to do then was run.

The rest of my thoughts pretty much agree with what's been already been said. Environment plays a big part in any Avatarverse battle.

I also wish to support this with this logic: If Katara can beat Azula, and Azula can beat Ty Lee, Katara can beat Ty Lee. If Katara can beat Mai, and Mai vs. Ty Lee is even, Katara can win. Finally, Katara and Sokka can beat Mai and Ty Lee, and if either Mai or Ty Lee could defeat Sokka, Katara could defeat the other, and since Mai can defeat sokka *breathes* Katara can beat Ty Lee.

Besides, Zuko can beat Mai and Ty Lee, so that means if Katara is of equal power to Zuko, then Katara can defeat Ty Lee.

I really don't think it's as simple as that. We're talking about firebending versus waterbending vs acrobatics vs knife throwing and so on. For the most part, each character's fighting style is fundamentally different. I sort of compare it to a game of rock, paper, scissors. Rock beats scissors and scissors beats paper, but that doesn't mean rock beats paper. Or (if you've played it), think of a rousing round of Super Smash Brothers Melee. Some characters are better suited to fight other characters because of differences in strength, speed, preference for melee or missile combat, attacks that pile on damage or attacks designed to push opponents away... The list goes on.

For me, it all depends on whether Ty Lee could get close enough to Katara to do any damage.
As we see in the season finally, Katara is very good at keeping her offense constant. For instance, Azula is an incredible fire bender, but Katara keeps her so busy that she doesn't actually get to do any full fledged attacks.

If Katara could keep Ty Lee at a certain distance, she'd win no problem, because although Ty Lee's great at hand to hand combat, she has no long ranged weapons.

On the other hand, if Ty Lee got close enough to work her magic, she could probably pwn Katara with one jab to her lower jaw (stomach point 5) to render her unconscious.

It's been said before but it pretty much all depends on the environment, the aggressor...and I don't think anyone's mentioned this one yet - how angry Katara is. She tends to be more powerful when enraged. ^_^