“A new pair of ” or “A pair of new”

Solution 1:

Both are found, but a new pair of . . . is four times as frequent as a pair of new . . . in the Corpus of Contemporary American English and over five times as frequent in the British National Corpus.

Solution 2:

The terms pair of scissors and pair of pants both refer to single objects that are, in normal use, never separated into the components that make them nominally a pair. When these terms are modified, they are modified as a whole — effectively a new pair-of-pants or new pair-of-scissors.

Shoes exist singly as well as in pairs (though rarely used individually). Did you see my other shoe? (But not, Did you see my other ∗scissor?) As such, the pair of shoes phrase is not locked as one.

I fully agree with other answers that new pair of shoes is more common than pair of new shoes. However, the latter is logical and correct, even though it is probably not the preferred form.

Solution 3:

In British English I am pretty sure that both usages would be correct but with different meanings.

A new pair of scissors implies that the pair is new, but not necessarily the scissors. For example, having tried to cut something and failed I might "try with a new pair of scissors". The scissors might be old.

By contrast a "pair of new scissors" implies the scissors are new (say newly bought at the shop).

EDIT

A better example (than scissors) would be the two example contexts:

"I spilled coffee on my leg, so I put on a new pair of trousers."

"I wanted to impress on my first day in the office, so I put on a pair of new trousers"

In the first example, the pair is new, in the sense that it isn't the pair (the "old pair") I was wearing when I spilled my coffee. But it is the pair that is new, the trousers could be old. In the second sense, it is the trousers that are new.

"Trousers" = "pants" in American English.