This is a very general question and difficult to answer. The kind of approach people will use is dependent on a lot of factors like the following:

- how much work can i spend on building my solver?
- what do i expect from my solutions (e.g. proven guarantees)?
- what kind of problem do i have (e.g. discrete search-space vs. continuous search-space and much more factors)

If i tackle a new problem, i'm a huge fan of the common techniques used in AI and OR community, at least for a prototype-solver which will be my first step. Most of the time one will be able to implement a solver in a very short amount of time (see above: factor one) which will perform not that bad (interesting example: link).
These (nearly) problem-independent solving-approaches include the following:

- SAT-solvers (mostly open-source-development-driven; famous example: minisat)
- SMT-solvers (basically sat with some extra-theories like real numbers)
- Mathematical Programming (the best solvers are the commercial ones like Gurobi and CPLEX, but there are open-source solvers too like GLPK):
- Linear Programming
- (Mixed) Integer Programming
- Quadratic Programming
- ...

- Constraint Programming (also many open-source solvers available: gecode, google or-tools))

Some comments about using these solvers:

- they are heavily used in practice (Scheduling, Vehicle routing...)
- you only need to model your problem (which is kind of abstract/declarative) and can trust the solver (especially the commercial ones)
- with more work to spend, there are numerous tunings to apply: especially the field of mathematical programming offers a lot regarding problem-dependent tunings (famous example: the concorde tsp solver is based on LP)

In theory, these approaches can handle any kind of np-complete problem, but of course there will be some differences dependent on the problem. Some hazardous simplified general statements:

decision-problem vs. optimization-problem:

- SAT/CP are really good at decision-problems
- Mathematical Programming is really good at optimization problems

discrete vs. continuous search-space:

- SAT is really good searching a discrete search-space
- Mathematical Programming favours a continuous one

solution characteristics:

- Mathematical Programming is good at bounding optimization-problems (in contrast to CP and particularly SAT)

Alternatively one can try to implement a problem-dependent algorithm with many common approaches like:

- Heuristics (highly problem-dependent) + Metaheuristics (more general, but need some problem-dependent heuristics internally ;in general these combinations are incomplete in contrast to the solvers above -> they may not find a solution even if there is one; additionally there are no guarantees)
- Approximation Algorithms
- ...

While tackling these problems one will learn that often these different approaches are somehow connected to each other (Mathematical Programming and Approximation Algorithms) or combined (MIP + CP; CP+Heuristics).

All in all: a difficult question!