The most common and perhaps the most serious objection to the insanity plea is that

Insanity plea stands for insanity defense, or also known as the mental disorder defense.  It is a defense by excuse in a criminal case, arguing that the defendant is not responsible for his or her actions due to an episodic or persistent psychiatric disease at the time of the criminal act.
The most common and perhaps the most serious objection to the insanity plea is that dangerous people go free.