1-1/4-inch-long drywall screw
The length of screw depends on the thickness of materials used. For hanging 3/8- to 5/8-inch-thick drywall on wood studs, 1 1/4-inch-long drywall screws will be sufficient.
If the same thickness of drywall is being installed on metal studs, only 1-inch long drywall screws are sufficient.
It is a good rule of thumb to always have a minimum of 5/8 inch of the drywall screw going into something secure like a wood stud, metal stud or furring strip. If you are screwing through multiple layers of material, measure the thickness of the layers, add 5/8 inch and the sum will identify the minimum size screw to use.
Fine-thread drywall screws
Drywall screws can be bought in coarse thread or fine thread. Typically coarse-thread screws are used in residential settings and fine thread screws are used in metal stud framing. In general, it is up to the drywall installer's preference.
Coarse-thread drywall screws have slightly greater pull-out strength than fine thread screws. Due to the fewer threads, they will install faster.
Fine-thread drywall screws have a sharper point which requires less force when starting the screw. Due to having more threads, they will install slower.