What are the three kinds of dangling modifiers?

There are actually a number of causes for dangling modifiers: some are dangling participles, dangling gerunds, dangling infinitives, and misplaced modifiers. In all cases, they result in sentences that are unintentionally funny.

My favorites are:

I heard it was going to rain on the radio. (Misplaced: "on the radio" should be next to "heard" unless you have a radio with a big rain cloud over it.)

Sandy was walking her dog in a really short skirt. (Misplaced: this sentence seems to say the dog was wearing a really short skirt, rather than Sandy.)

Having broken its leg, we took the dog to the hospital. (Dangling gerund-- did WE break the dog's leg?)

A clean coffee pot is necessary to enjoy a good cup of coffee. (Dangling infinitive-- where's the subject in this sentence? Who is drinking the coffee-- the coffee pot?)