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I have read that one should not leave greasy fingerprints on their spark plugs or on their light bulbs. Why is this a thing? On what else one should not leave fingerprints? How does one avoid leaving fingerprints (latex gloves or, say, cleaning the light bulb after putting it in)? How do you clean your fingerprints if you leave any?

Why is this a thing?

For bulbs it is - spark plugs don't care. The problem with getting fingerprints on bulbs (specifically halogen-type bulbs) is actually the grease that gets left behind - the bulb will get quite hot when lit and grease on a portion of the bulb will cause differences in the rate of heating on the bulb surface and thanks to the way materials expand when heated this means you get different parts of the bulb expanding at different rates and... crack.

Where the bulbs are made from quartz glass (such as in certain more powerful bulbs from some manufacturers) there is also the issue of devitrification - where the oils/salts left behind in the finger print can potentially cause the material to weaken although typically car bulbs aren't going to get hot enough for this to occur although some fogging of the glass may occur.

On what else one should not leave finger prints?

Probably a bit broad to answer here I'm afraid.

How does one avoid leaving finger prints (latex gloves or cleaning say the light bulb after putting it in)?

Yep clean latex gloves (cotton gloves are even better if you have them - but latex or nitrile works just fine) or a holding the bulb in a clean cloth works.

How do you clean your finger prints if you leave any?

You can clean them afterwards as well, a good wipe down with a clean microfibre cloth will do the job.

Fingerprints on spark plugs aren't a thing, the natural oils on your skin won't affect their operation. If, however, your fingers are covered in car grease or dirt when you handle the plugs then that can cause issues, either interfering with the electrode or the cable connection to the plug terminal.

If you foul the plugs with dirt or grease then clean them before use. Any dirt on the electrode is going to end up in your cylinder!

For halogen lights @Motosubatsu's answer is pretty comprehensive, I would only add to be careful if handling with facial tissues (i.e. Kleenex), which some people do. Some types of tissues have been treated with moisturizers which will contaminate the bulb and cause the same issues as fingerprints. Toilet tissue generally doesn't, neither do paper towels.

Despite the two previous answers stating that fingerprints on spark plugs are not a problem, I'll mention that they are.

I have personally replaced plug wires that showed traces of carbon where the spark was "leaking" around the plug due to grease on the plug ceramic. This is especially apparent on aftermarket ignition systems with higher voltage and lightly-coloured (orange) plug wires. Granted these were vehicles modified for competition twenty years ago, however today's ignition systems are on par with those for voltage.