Pour rubbing alcohol into a small container. Dip a cotton swab into the rubbing alcohol and rub the permanent marker stains until they are no longer visible. Wipe the desk clean with a damp cloth. Continue if the marker is still visible.
Apply hair spray directly to the permanent marker spots. Allow the hair spray to soak into the spot for a few seconds. Wipe the area with a soft cloth. If necessary, apply more spray to the spots. Continue to step 3 if the permanent marker is still present.
Add a few drops of nail polish remover to a cotton ball. Gently rub the permanent marker spots with the cotton ball until the marks are gone. Wipe the desk clean with a damp cloth. Continue to the remaining steps if the permanent marker is still visible on the melamine desk.
Spray a water-displacing spray such as WD-40 directly onto the marks. Rub the spots and spray with a soft cloth. Apply more water-displacing spray and continue rubbing until you have removed the permanent marker completely. Wipe the desk clean with a damp cloth.