Weekly, Daily, Hourly --- Minutely...?

"Every minute" or "once a minute" is correct. There isn't a single word for this. RegDwight points out "minute by minute" and "on a minute-by-minute basis", but note that "on a minute-by-minute basis" means something different from "every minute" - "he updates his blog on a minute-by-minute basis" - he updates his blog very frequently, perhaps not actually every minute - as opposed to "he updates his blog every minute" - he actually updates his blog every single minute.

Communication is the goal of using language. Any word or even non-word that communicates what you intend to the people who read it or hear it, may suffice. Within that broad stroke, the details are that some words may communicate 'better' in some way. What is 'better' may be more efficient, more accurate, more pleasing, or in some other way 'better'.

A (not the) "correct" word for "happening every minute" is 'minutely'. In both US and British English, the word is pronounced /ˈmɪnᵻtli/, where /ɪ/ is the vowel sound in 'kit', "ᵻ represents free variation between /ɪ/ and /ə/", and /i/ represents the vowel sound at the end of 'happy' (OED).

The adjective and adverb 'minutely', in the given sense, is described variously by dictionaries. In OED entries from the Third Edition, updated March, 2002, the word is described as "rare". It is given two out of eight for the adjective frequency, three of eight for the adverb, both in contrast to four of eight for the adverb in the sense of "on a small scale". In Collins, the word frequency is given as two out of five, exactly the same frequency shown for 'minutely' in the sense of "done in great detail". American Heritage (online) does not define the adjective or adverb in the given sense. Dictionary.com (based on Random House Unabridged) makes no observation about the frequency of use.

That none of the four dictionaries consulted describe 'minutely' as archaic, in the sense of "minute-to-minute" or "minute-by-minute", is not in itself a conclusive answer to the question of whether it is archaic or not. My observation, however, is that 'minutely' is not archaic, or if it is archaic, it is not primarily so. Rather the word is infrequently used, and may be perceived as "belonging to an earlier period". Yet the semantics of 'minutely' (/ˈmɪnᵻtli/) are not archaic, nor is the form; the only remaining potential dimension of archaism is chronological, in that it may be perceived, rightly or wrongly, as having been used more commonly in the past than it is used now.

There remains a question of whether, supposing that 'minutely' in the sense of "minute-to-minute" is perceived as archaic, that perception is sufficient reason not to use the word in that sense. The logic against such use may be circular and so not compelling. For example, the logic may be that because 'minutely' is less used now than it was in the past, it should be used less now and in the future. If language operated on that principle, rather than the principle stated in the first paragraph of this answer, words and particular senses of words would not fall out of, then return to, common use.

It can easily be observed, though, that words themselves as well as particular senses of words do fall out of then return to more frequent use. Other influences than the presence or absence of archaism affect that process.

Finally, arguing against the use of 'minutely' in the sense of 'minute-to-minute' because it is archaic flies in the face of the sense of 'archaic', when accurately used of language:

b. esp. of language: Belonging to an earlier period, no longer in common use, though still retained either by individuals, or generally, for special purposes, poetical, liturgical, etc. Thus the pronunciation obleege is archaic in the first case; the pronoun thou in the second.

OED

The use of 'minutely' in the sense of "minute-to-minute" or "minute-by-minute" may well be a general use "for special purposes", for example, technical purposes. Like thou, 'minutely' is still used. It is retained for general technical use.

OED, in the entries updated in 2002, attests adverbial use of 'minutely' in the given sense as recently as 1962, and adjectival use as recently as 1994. Further, it does not take much searching to uncover technical uses of 'minutely' in the given sense in 2018; for example, a search using Google Scholar to find instances of 'minutely data' returns 2490 results from 2018 as of today (31aug2018).

How about "per-minute"? I'm working on a database design and have a frequency field that is fixed to the following set of possible values: "per-minute", "hourly", "daily", "weekly", "monthly", "quarterly" or "yearly".

So far "per-minute" is the shortest and most sensible approach I've found.

"Minutely" seems to have another meaning:

With great attention to detail; meticulously.

and all the examples seems to fit this meaning, rather than the 'minute by minute' meaning:

minutely examples in sentences