Throughout the film, several characters use the word "alibi" in a wrong sense. They use it to refer to the argument that Catherine would not have committed a crime very similar to one described in a novel of hers. But that is not what the word alibi means. It refers to a piece of evidence which shows that a suspect could not have been at the scene of a crime at the time of that crime (the literal translation of the Latin word "alibi" is "elsewhere"). The description of the ice pick murder in Catherine's novel does not prove anything of that sort.
When Catherine arrives to be interviewed there was confusion as to how and why she waived her right to an attorney, but Catherine, Gus, and Nick should've have realized she waived her right to an attorney when she said to them "why would I need an attorney?" as they were leaving her home.
When Nick Curran and Gus arrive outside and enter Johnny Boz's home at the start of the film, Nick's hairstyle is combed and wavy as he goes upstairs to Boz's murder scene. When Curran walks through Boz's bedroom to examine the crime, his hair is gelled and slicked back.
After Catherine scratches Nick, drawing lines of blood, she flips him over and is on top of him, theoretically smearing that blood all over the pristine white sheets. No blood is ever visible on the bedding.
When Nick gets up from bed, the scratches on his back are barely visible. When you see his back reflected in the mirror when he speaks with Roxy, the marks are bright red, bloody. When he returns to bed, they're the way they were in the previous scene.
The police would have requested a DNA sample from Catherine, as this would have proved she was with Johnny Boz when he was murdered. DNA has been used in criminal investigations since 1984. DNA evidence was used to secure a conviction in a rape case in Florida in 1986, and in January 1988 Colin Pitchfork was convicted of murder by only using DNA evidence.
A scene where Nick is at Catherine's to question her about Hazel Dobkins, he is looking through news clippings on himself on the accidental shootings when he was on vice. When Catherine reveals to him she is using him as a reference for the Detective in her novel he comes across a newspaper clipping headlined "Grand Jury Probe Continues" with a his picture in a specific pose to the camera. He displays the same exact (or eerily similar) pose in the next shot when he reacting to this news.
The goof items below may give away important plot points.
When Nick arrives at his apartment after Nilsen is killed and Catherine is waiting for him. They go upstairs and have drinks. Nick starts chopping the ice into a few chunks then Catherine destroys the entire block of ice. Now the ice in their "Rocks" glasses are quite large until Catherine walks across the room and the ice is all gone (ice doesn't melt that fast, even if the bourbon was heated). Plus the liquid in the glasses got much darker, this would never happen because the water would have lighten the liquid in the glass, not darken it.
In reality the police would have simply got the court to request that Catherine Trammell provide a DNA sample, since she was the last person seen with Johnny Boz. Since her bodily fluids and skin fragments would have been all over the bed this would have been enough to convict her for his murder.
When Lt. Nilsen is found dead of a close contact gunshot wound to the head, Gus opines that the weapon used was a .38 caliber revolver. Lt. Walker then demands that Nick turn over his weapon. Nick produces a Glock semi-automatic pistol and Lt. Walker smells it, presumably to see if it had recently been fired. Although the bullet diameter could be said to be similar and thus produce similar bullet holes (a point much too esoteric for this film), Nick's pistol is definitely not a revolver.
The blonde woman murderer in the opening scene has a clearly visible scar on her left shoulder. Later in the film, the same scar can be seen on Catherine Tramell's left shoulder. This gives away the whole story, since the point of the film is to figure out who the killer is.