While reading the D&d 5E Basic Rules, I noticed that under "Other Dexterity Checks", it said "Play a stringed instrument". This led me to wonder what ability is used for playing other types of instruments- specifically, woodwind instruments (it seems that woodwinds and strings make up 80-90% instruments listed in the 'Tools' section of Chapter 5). I searched everywhere, but couldn't find anything.
The things listed under each ability are just examples, not a complete and exhaustive list of the only things possible with ability checks. They're listed to give an idea of what generally falls under each ability, and provide exemplars that DMs can extrapolate from to unlisted situations encountered during play. Based on the things listed under each ability, playing a woodwind—or any kind of instrument—is also a Dexterity check.
I could see an exception for an instrument that required no manual dexterity to play right but did require massive lung capacity, and ruling that a "performance" with that instrument was a Con check. But such unusual circumstances are precisely why the abilities are described using examples, and the exact check required is then left to the DM to decide based on that guidance and their own judgement.
Woodwind musical instruments are a tool (page 50 of the Basic D&D Rules (v.01)), and thus are not assigned to any specific Ability to use.
Page 49 of Basic rules (v.01)
Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver’s tools, or a Strength check to make something out of particularly hard wood.
However, even if a woodwind instrument was a skill not tied to any specific tool, you could still use any relevant ability score for related checks.
Page 59 of Basic rules (v.01)
Variant: Skills with Different Abilities
Normally, your proficiency in a skill applies only to a specific kind of ability check. Proficiency in Athletics, for example, usually applies to Strength checks. In some situations, though, your proficiency might reasonably apply to a different kind of check.
That being said, I would advise that woodwind instrument either uses dexterity, charisma, or possibly even constitution depending on what type of song you are trying to perform with the instrument. Trying to play a fantasy version of "peter and the wolf"? That might be dexterity or intelligence if you are reading from sheet music. Are you trying to distract a mob or an angry room in a tavern? You should probably use charisma. Trying to do a contest of who can play the instrument the loudest or longest? I'd go with a constitution check. But in the end, you use the check that makes the most sense to the table.
I'm going to look at this one from a slightly real-world point of view, as I'm a woodwind (and string) player.
If you want to simulate this really properly, it's going to depend on what the instrument actually is. My main instrument is the recorder, which is easy to play from a physical point of view (although the contrabass requires quite a lot of puff), but requires high dexterity in the fingers and tongue and good diaphragm control to play it well. Therefore I'd probably say that the primary stat for a recorder is dexterity.
That's going to apply to a lot of instruments - transverse flute, clarinet, oboe... reed instruments take more pressure to play, but that's only something to factor in if playing for a long time - I think a recorder player might be able to go longer than an oboist without needing to do a constitution check to see if they're too tired to continue (or too tired to carry on playing well).
If, on the other hand, they've decided to play the bombard you're going to need them to check constitution after five minutes or less. Those things are hard work.
Finally, if you want a character to play an instrument really well they're going to have to go away and learn how first, so anything that actually does involve the performance of music is going to have to have a skill component as well to represent a performer's musical education. This may or may not matter to your particular campaign, of course.
TL;DR: dexterity is the primary attribute for most instruments. Constitution comes in if playing for a long time, which can be very tiring, and different instruments tire you out at different speeds.
Strength may be required for picking the thing up, if they've decided to take up the sousaphone.