A roller makes the job go faster, but it prevents you from using paint with a glossy finish. A brush can use any paint, but it takes longer and can leave visible brush strokes.
The best solution is to use the roller on the large surface spaces of the cabinets, then follow with the brush in areas where the roller won't reach, such as narrow edges.
"Use a good quality smooth mini roller to give the best finish [... and] a sustainably-sourced paintbrush for all the edges and corners," says Victoria Yardley of Victory Colours.
If you have a lot of surface space on the cabinets, a 6-inch roller should deliver the best results. For smaller cabinets, stick to those measuring 4 inches or less in length.
If you are painting on larger cabinets, select a larger brush to complete the job faster, and use one with angled bristles if you plan to cut into the corners inside the cabinet.
Think about the type of paint you plan to use, as well. Latex work best with polyester or nylon bristles, while oil-based paints work better with natural bristles.