Regular old Kalman Filters are the best (literally perfect) when your problem fits all their requirements. They also have a lot of nice properties if you're dealing with a problem that mostly fits their requirements. But the linear-gaussian requirement is pretty steep, they don't always work.
I don't like the EKF much and prefer the UKF. The core filtering code is a little more complex but they're much easier to actually work with; you can give them arbitrary functions like a particle filter.
Particle filters have the advantage of being able to handle arbitrarily wacky distributions. But they are random and do some wacky things in edge cases. They'll behave much more poorly in low-evidence situations than other filters will. And they fall over spectacularly if you switch from low-evidence to high-evidence (there's a workaround for this but it's still counterintuitive). Finally they're just more computationally expensive than the others.
I don't like the EKF much and prefer the UKF. The core filtering code is a little more complex but they're much easier to actually work with; you can give them arbitrary functions like a particle filter.
Particle filters have the advantage of being able to handle arbitrarily wacky distributions. But they are random and do some wacky things in edge cases. They'll behave much more poorly in low-evidence situations than other filters will. And they fall over spectacularly if you switch from low-evidence to high-evidence (there's a workaround for this but it's still counterintuitive). Finally they're just more computationally expensive than the others.
Birch sounds interesting, I'll take a look.