DIAGNOSIS Smearing happens when you're using a dry cloth on a screen that has oils from your fingers or dust that's slightly greasy. You're just moving the grime around. You need the right cleaner and technique.
FIX First, power off the monitor. Unplug it if you want to be extra safe. 1) Use a dry, clean microfiber cloth (like the kind for glasses) to gently dust the screen in straight strokes. Don't press hard. 2) If smudges remain, dampen a second microfiber cloth with distilled water. Do not spray anything directly onto the screen. Lightly dampen the cloth, wring it out completely so it's just barely moist. 3) Wipe the screen in one direction, gently, using minimal pressure. 4) Immediately dry/buff the area with the dry microfiber cloth to prevent streaks.
IF THAT DIDN'T WORK For stubborn spots, you can use a 50/50 mix of distilled white vinegar and distilled water on your damp cloth. The mild acidity can cut through tougher residues. Again, apply to the cloth, not the screen. Test a small, inconspicuous area at the very edge first.
HEADS UP Never use window cleaner, alcohol, ammonia, or abrasive cleaners. They will strip the anti-glare coating right off the screen, leaving permanent, cloudy damage. Also, avoid paper towels or tissues—they contain wood fibers that can microscratch the soft screen surface.