Can 'nook' be used outside the saying 'nook and cranny'? Is it common?

Solution 1:

The British National Corpus contains 75 instances of nook (I haven't checked nooks), although they're not all independent. E.g. three of the uses in a name are references to the same play, Rookery Nook.

Of those 75:

• 25 include the word cranny in close proximity.
• 17 are in names of houses, pubs, plays, streets, etc.
• 8 are in poetry (which I categorise apart because it tends to use rare words to fit metre and rhyme).
• 24 fit none of the above categories, so we could call them "general usage".
• And since that adds up to 74, I've lost one while I was sorting them into categories.

The Corpus of Contemporary American English has 1088 instances of nook, of which 208 have the word cranny within four words. (I didn't count these manually, but used the collocates feature of the search).

So in both corpora the word nook occurs more commonly without cranny than with.

As to commonality in general, the full frequency distribution of COCA is not available without payment, but the top two lemmata in the sample data are

Position  Word      Frequency
7309      attic     2711
17311     tearful   542

so it is maybe around the 12000th most common word in the corpus, and almost exactly twice as frequent as the perfectly common word tearful.

Solution 2:

Yes, you can use nook without the cranny, for nook merely means "a corner or recess, especially one offering seclusion or security." (Oxford Dictionary) See here for sentence examples of nook by itself. As to whether it is common, that is something I wouldn't concern myself with. Commonality, or lack thereof, should not dictate whether something is used or not. Usage of language should be dictated only by communicative qualities; that is, whether the term or word in question adequately and succinctly communicates an idea, regardless of how old or common it is. This is just my opinion, however.

INCIDENTAL ADDENDUM:
My answer and I have been criticized by some, as you can see in the comments, so allow me to clarify: I am speaking solely about words that are in current use and are not considered “obsolete.” I do not think that one should use just any word; one still has to take the time to study the nuances of words. But people are too swayed by how common a word is as opposed to how expressive it is. Language’s sole purpose is for communicating, after all.

Solution 3:

Out of curiosity, I searched both of Tolkien's The Hobbit and The Lord of the Rings for "nook":

[The cave] seemed quite a fair size, but not too large and mysterious. It had a dry floor and some comfortable nooks. At one end there was room for the ponies; and there they stood (mighty glad of the change) steaming, and champing in their nosebags.

^{The Hobbit, Chapter 4: Over Hill And Under Hill, p.53}

A little later Frodo came out of the study to see how things were going on, and found [Lobelia] still about the place, investigating nooks and corners, and tapping the floors.

^{The Fellowship of the Rings, Chapter 1: A Long-Expected Party, p. 57}

The westering sun was caught into clouds, and night came swiftly. They slept as well as they could for the cold, turn and turn about, in a nook among great jagged pinnacles of weathered rock; at least they were sheltered from the easterly wind.

^{The Two Towers, Book Two, Chapter 1: The Taming of Sméagol, p. 227}

‘We had better try a way back southwards along the line of the cliff, I think,’ said Sam. ‘We might find some nook there, or even a cave or something.’ [...] They did not find the going any easier at the broken feet of the Emyn Muil. Nor did Sam find any nook or hollow to shelter in: only bare stony slopes frowned over by the cliff, which now rose again, higher and more sheer as they went back.

^{ibid., p. 237}

Interestingly, nook and cranny doesn't appear at all. As to whether Tolkien's use of nook is "idiomatic", "appropriate" or "common", I'm not sure. I can say however that you'd be in good company.

Solution 4:

Yes 'nook' without 'cranny' is fine. From OLD:

Nook: a corner or recess, especially one offering seclusion or security.

My cat has currently found himself a comfortable nook in the garden to sleep in.

And, most importantly, the term feels perfectly normal, not unusual or archaic to me, a native English speaker. That's the only real test.