Halogen and LEDs are the most common lamp types in modern microscopes. The color spectrum of LEDs is more narrow than that of halogen lamps and the light intensity is generally still lower in LEDs, which I expect to change in the future. Both tungsten and halogen bulbs are similar in that both of them have a tungsten filament which generates the light. Halogen lamps are filled with a halogen gas, which lengthens the life of the bulb.
The wider color spectrum of halogen lamps, in comparison to LEDs, is a certain advantage for some stained specimens. Apparently the colors are rendered more naturally in halogen light. The disadvantage of halogen bulbs are that they do generate heat and that the microscope must be plugged in into a power source. LED microscopes may also be battery operated, which is an advantage in classrooms where there is no power available for each table.