Most airplanes only take only the amount of fuel with them that is needed for the planned flight (plus a bit more for unexpected wind changes etc.). Usually, the tanks are mostly empty at the landing in any case.
Depends on how you define 'mostly empty'.
The minimum fuel quantity for IFR flights at reaching the destination is a reserve for 45 minutes cruise + fuel to reach an alternate airport (in case the weather is going to be bad, but most flights today carry alternate fuel anyway) + minimum fuel that has to remain within the tanks because of design reasons (tanks are never fully empty unless there is a general airplane check). So a Boeing 727 for example still has at least about 7300 pounds of fuel in its tanks at landing (without alternate fuel!), which is almost 15% of the total fuel quantity.
This has nothing to do with the landing fuel, but beside all that fuel loaded before flight, taxi fuel also is an extra fuel load, which even allows to exceed the maximum take off weight during taxi out.
For the fuel dumping speculations: the whole Boeing 737 family has no fuel dumping capability. Fuel dumping only is used on heavy airplanes because of their long landing distances.