You have two options --
John listening to the radio is an everyday occurrence.
John's listening to the radio is an everyday occurrence.
The first has John as the subject, and the second has listening as the subject. In the first sentence, "listening to the radio" modifies the subject John, and in the second, "John's" modifies the subject "listening to the radio". As Implode said, these modifiers shouldn't really be integral to the sentences, and the sentence should make a lick of sense without them. So let's take them out:
John is an everyday occurrence.
Listening to the radio is an everyday occurrence.
Obviously, the second is preferable.
Except it's not actually an extra bit of information. It's the point of the sentence.
If it's the point of the sentence, it should be the subject.
Close, but not quite. You're thinking of a prepositional phrase. A prepositional phrase can always be removed and not modify the structure of a sentence. The subject remains the subject, the verb remains the verb, the object remains the object.
In this case, however, the two sentences are actually equivalent. "John listening to the radio" is what is occurring. It happens every day. It is an everyday occurence.
You can also say that "listening to the radio" is what is occurring. "Listening to the radio is an everyday occurrence" is a perfectly valid statement. In this case, however, the fact that John is doing it is the extra information. It is
John's listening to the radio, but
listening to the radio is in fact the subject.
Still don't believe me? Consider a hypothethical conversation taking place in the next room.
A: What's going on in there? I hear music.
B: John's listening to the radio. He does it every day.
Or consider this:
A: What is John doing in there?
B: Listening to the radio. He does it every day.
The apostrophe denotes possession, not contraction. It's not "John is listening to the radio". John "owns" the action of listening to the radio, and that act is the subject of the sentence.