What training method works best for your dog?

I'll answer the question vaguely, because the forum rules enforce/encourage? a certain level of self-censorship .
I use mostly positive reinforcement, clicker style, luring, shaping. I can't think of any dog that wouldn't benefit from motivational training, but while I will problem solve using various forms of "management"--crates, doors, baby-gates, leashes, long-lines, fences, I do recognize that "management" can also be aversive and I'll balance that with limited use of less politically correct aversives.
A leash and flat collar can be plenty aversive if the dog can't run, jump & play. Even long-lines are cursedly frustrating for a fast dog.
Favourite trainer, Michael Ellis, and much of what he has to say is about the power of motivation, food, tugs, toy.
And a great hint, re: loose leash walking skills, is this and works for those who would never use certain tools. When training, clip the lead to your dogs training collar, if that's a flat collar, that's the one. Your in the mood, do all your things, click and treat, red light green light, etc.. If you run out of time, or patience, or for whatever reason need to lower your criteria, switch the clip to a harness. That way your dog never gets to practice pulling, lunging, whatever, on their flat collar, and you can preserve your sanity by not setting yourself and your dog up for failure.
I use a harness for long-line work, because the criteria for that is much lower.

Like Cookieface and Artdog, I am primarily a reward based trainer. With my personal dogs, I tend to go with shaping and targeting most often to train new behaviors. I use capturing fairly frequently, especially for default behaviors. I really enjoy chaining/backchaining behaviors. I also have no problem with using luring, if that will get me what I want the quickest.

I avoid modeling (often aversive to the dog and just not the quickest results) as well as physical/verbal corrections, leash corrections, etc.

I rely on management first and foremost to prevent unwanted behavior and reinforcement for such behavior.

I also do not claim to use only R+. Really don't think anyone can. If my dogs do something I don't want them to, I'm not going to stand by an watch or ignore (unless it is attention seeking behavior like demand barking or puppy jumping/mouthing). I will address it, but in a non-confrontational way, generally calmly and quietly interrupting and redirecting to a different activity or behavior I do find appropriate. Very often during a training session for a new behavior or criteria, I just withhold the click and treat when my dogs offer something that was not correct, causing my dogs to try again (not at all a big deal as they are clicker savvy and have a lot of shaping experience).

Other Consequences (normally formal training/sports working with well known behaviors and distractions) would include me stepping out of heel or even walking a little ways away, giving the reward the dog would have earned to a different dog, or ending training mid session (strongest correction I intentionally give and it is used incredibly sparingly - as in can count on one hand the times I have done this in 2.5years with my sport guy- my dog would have had to have done something extreme). Typically the first two (stepping away or giving reward away, are done in setups where a distraction is presented when working on something well known. Success is rewarded. Wrong choice results in no reinforcement from distraction (is a person, he ignores the dog, if food, toy, etc. a helper eliminates access) and a consequence (moving away or giving away reward for experienced dogs. For dogs newer to this we don't leave or give away reward, just wait, let them figure out there is not reward for leaving, and verbally support when they make the decision to return. Basically the few corrections I give are P- and largely due to need to make sure my dogs are not rewarded for the unwanted behavior when they make a mistake. Also tend to be quite effective as they love working because of the insane amount of reinforcement they have earned. The lack of or loss of reinforcement speaks volumes for a dog truly savvy to reward based training. No verbal, physical, or leash correction needed.

That said, I try very hard to do what I can to set them up to succeed. Unfortunately at a certain point especially when working on distractions, there is a right choice and a wrong. I build gradually so there are far more successes than failures, insanely so. And if my dogs fail 2-3 times in a row, I understand they are not ready for that stage yet. I change criteria so that there is a success to end on and then set up a new plan to address that particular scenario successfully.

I would say I probably have "smart" (although I would probably use "easily motivated") dogs, an Aussie and a Border Collie. I am very similar to kmes, shaping and targeting are my go to training methods, but I would say I am a science-based trainer, and try to operate as +R as I possibly can. I definitely use capturing for cute behaviors, and I try not to lure, but I usually do with young puppies.
I am also not permissive, my dogs don't get to do anything they want, but we have a good relationship, so any corrections are usually a minor voice correction of "Oops!", but I'm in the process of trying to phase out voice corrections all together. I usually withhold the reward if my dogs are not offering the behavior I want, or if just one if performing, they get both rewards.

I would never assume/ask my dog to perform for me out of "respect". If I were to ask that of my dogs, it's only fair that I go to work and do my job out of "respect" for my boss, and nothing more. Yeah, I don't think so. I love my job, and really like my boss, but a girl's gotta eat.