Sim's response to a marriage proposal will be random and not based on relationship level.
Insane Sims are not affected by ghosts.
Insane Sims can talk to themselves to fill both the social and fun motives, alleviating both the Lonely and Stressed Out moodlet.
Insane Sims can create distorted paintings using Paint Stylized Still-Life.
Insane Sims will autonomously sit on a toilet or in a bathtub without using it.
The Speak Madness social interaction will creep-out non-insane Sims but will entertain other Sims with the Insane trait, and give out a bonus in relationship the first time if the Sims have not found each others' Insane Trait (and mark it as discovered). It takes a very high relationship, presumably best friend level, to get a sane Sim to react positively to the said interaction.
Like Childish Sims, an Insane Sim can gain fishing skill while Fishing in the pool. However they will not catch any fish.
Insane Sims autonomously consume fruits from the inventory even when not hungry, often yielding the Stuffed moodlet.
Insane Sims may exercise and/or go on computers while at parties, giving the "Rude Guest" moodlet to the host, unless the Insane Sim in question has the Inappropriate Trait.
Insane Sims may get the wish simply to go out at random.
With Pets installed, Insane Sims may sometimes act like one of the major pet animals.
There is a glitch with the Insane trait. With Pets, an insane child may start stretching because there is no animation for kids doing the "Acting like a horse".
Interestingly, the Talk about conspiracies interaction will be considered friendly, and it will usually be received positively. On a side note, the Catch anything? interaction will allow the Sim to discover if the target Sim has the Angler trait, and if so, give a bonus to the relationship.
When swimming or exercising, Sims may choose other clothes types.
If Sims have both the Insane and Never Nude trait, they may take showers and baths in their everyday clothes, formal wear, etc.
If Supernatural is installed, Sims with the insane trait will get the "Lunacy" moodlet during a full moon.
Insane Witches may autonomously cast ice blast on plumbing, breaking them in the process.