Of course, there are two problems with these methods: you can still be identified based on the pattern of scarring (like John Dillinger!), and in most cases your fingerprints will eventually grow back. They grow back because your fingerprints are determined at the genetic level.
Unless you have a mutation in your SMARCAD1 gene, that is!
An incredibly small percentage of the population is born without fingerprints, a condition known as adermatoglyphia. These people have eerily smooth fingertips, which starts with a mutation in the SMARCAD1 gene, and relies on a long chain of events in the womb to be expressed.
The genetic link was provided by a Swiss family where half of the family members are born with adermatoglyphia. Researchers from Tel Aviv studied the family's genetics extensively before being able to identify the gene that caused the issue.
Scientists don't yet understand all of the biological processes which rely on adermatoglyphia, but I imagine there are some less-than-ethical doctors who are pondering the question. Imagine how much money you could make if you were able to offer a successful "fingerprint removal" genetic surgery!
Sure, your new fingerprint-less fingers would still be identifiable. But you would not be able to be connected to your past crimes beyond a reasonable doubt. (Not on fingerprint evidence, at least. Our increasing reliance on DNA evidence would no doubt make this procedure a moot issue.)
Fingerprints typically develop in the womb, between 6 and 13 weeks. Your fingerprints remain the same throughout life, the only difference being that they grow larger as you age from a child to an adult. Fingerprints are remarkably persistent, and according to this forensics lab, they are "one of the last features to decompose after death."
Fingerprints have been used as unique identifiers since ancient times. They were used as signatures in ancient Babylon in the second millennium BCE. They were also used as signatures in ancient China as early as 246 BCE. Fingerprints have been taken by the police to help solve crimes since the time of Hammurabi, who lived from 1792 to 1750 BCE.