As your question is somewhat ambiguous since the "sentence" is really everything that has been written, let's first address the part that you have written in capital letters: "It occurred to her that there are really no new styles." This part of the entire sentence written above is both an independent clause and a relative (dependent) clause.
Now, an independent clause is one that can stand as a complete thought or sentence by itself; that is, an independent clause contains a subject and a predicate that on their own make logical sense. "It [subject] occurred [predicate] to her" makes sense on its own, (although the reader does need a little more information since the he/she does not know what "it" is). Therefore, this is an independent clause.
A relative clause is one that begins with a relative pronoun such as that, which, who, whom, or whose. A relative clause is also called an adjective clause since it modifies a noun or a pronoun. In this case "that there are no new styles" modifies the subject pronoun It. This clause is a dependent clause because it cannot stand on its own and have meaning. Also, the first part of the sentence, "When Jane went shopping" is a dependent clause, an adverbial clause that modifies the verb occurred.