When Jane went shopping, IT OCCURRED TO HER THAT THERE ARE REALLY NO NEW STYLES. Is this sentence an Independent clause or phrase or dependent...

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.