On their own, little to nothing of value. However, as a unique identifier of an individual, it can help identify who was in the vicinity of the body. This might not be the real killer, but Locard's Exchange Principle, which states "Every contact leaves a trace."
Consider a case where a deceased man is found in a room that can only be accessed by four people (The Victim, the Co-Worker, The Boss, and The Janitor). While investigating the crime scene, the investigative team will take finger print samples of the three living witnesses and the deceased body. They also dust various objects for prints and find five unique prints... since only four people can enter the room, the fifth set may point to the perpetrator of the murder... or it could be that the man was showing his new girlfriend a hot time and she left him very much alive.
In this situation, they offer little to solve the crime without further evidence, but they can point in the direction of a new person who accessed the room despite the boss's claim that no one other than the four people named entered the room. It is out of the ordinary and will be looked at.
As for using the palm to handle the object, your "Palm Print" is also unique. As are the toes and sole of your foot, all of which can be used to link back to likely one person. In 2015, it was discovered that identifying someone's sex by a fingerprint is even possible, so they can even eliminate the number of likely killers by half.
The use of unique body attributes is called bio-metrics. Fingerprints are the most commonly used in law enforcement and thus form the largest database of bio-metrics in the world. It is so commonly used that, unique identifiers for other metrics, even those with no human body relationship, are also called "fingerprinting" because the strength of human finger prints is so strongly known in the public mind (I.E. DNA Fingerprinting is using DNA evidence in a similar manner).
It should be noted that most "Cheap" finger print scanners are not reliable and can be fooled 80% of the time.