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)
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)
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.