When Monk is about to accidentally erase the files, the computer page says "1998 to present" but later, when terminating Monk, the Commissioner later refers to the fact that "ten years worth of files" were erased. 1998 to 2004 is six years, not ten.
In the coroner's office, Monk looks at the torso and says that the pox vaccination indicates that the victim must be from the Baltic States and is at least 26 years old because the pox vaccination ended in 1978. In Estonia, they still had the pox vaccination in 1988 and maybe in the other Baltic States as well. So Monk's conclusions are slightly off.
At the very beginning, a splash is heard and a woman is seen swimming underwater in a pool. However, there are no circular ripples spreading out over the surface of the water indicating the recent disturbance of her entering the water.
When Monk examines the torso, he determines that the fiancé is a left-handed mountain climber. However, in every scene where Paul Harley appears, he wears his watch on his left hand. Generally people who are left handed wear their watch on their right hand, and vice-versa. However, it is improbable, but not impossible, to find left-handed people who wear their watch on their left hand.
The goof item below may give away important plot points.
Solving this case revolved around the fact the murder victim had previously cut her long hair off and sold it to a wig manufacturer. The hair was used to make a toupee and the police used the DNA match between the toupee and murder victim to solve the case. The problem is is that a DNA sample can only be made from the root of hair and not the hair shaft. Since the hair was cut and not ripped out it would be impossible to make a DNA match.