I, personally, have no desire to have a tattoo either in Valheim or real life.

For those who would like to have tattoos in Valheim, I would like to make the following points:

Given the nature of a tattoo, it should not be removable. Tattoos only became removable without cutting off the skin on which they were placed in my lifetime. Vikings would not be able to remove tattoos without removing the skin.

Simply painting one's face or other body parts might also be included. Such decorations would be removed by immersion in water or running around in a rain storm.

A tattoo or a face painting should require new tools. Knives and fingers might suffice for large and crude tattoos and face paintings but more elaborate designs should require needles and brushes. A solo Viking would also require some sort of mirror for a facial tattoo.