It is known that the halting problem cannot have a definite solution, one that a) returns true <==> the program does indeed halt, and b) handles any input, but I was wondering if there are good enough solutions to the problem, ones that can maybe handle certain types of program flows perfectly, or are able to identify when it cannot correctly solve the problem, or one that is right a high percentage of times, and so on....
If so, how good are they, and what ideas/limitations do they rely on?
The normal approach is to constrain program behavior to an effectively calculable algorithm. For example, the simply typed lambda calculus can be used to determine that an algorithm always halts. This means that the simply typed lambda calculus is not Turing complete, but it is still powerful enough to represent many interesting algorithms.