These 3 species' songs shouldn't give too much trouble, but they do occasionally. Generally, it is the male birds that sing but the females do occasionally, though their song is usually weaker.
The Chaffinch has a short, fast rattling song, best remembered by imagining a cricket bowler running up to the crease with the ball and then delivering with a flourish. Alternatively, the song is usually rendered as "chip chip chip chip chip - chooipchyoo". Go on - try it! The song is usually delivered from a perch from February until June and lasts for 2 or 3 seconds, but they will often sing it 5 or 6 times a minute and may continue doing so for a very long time!
The Greenfinch sings either from a perch or in flight and it has a song for each location! One is the wheezy flight song which is quite distinguishable (Example 2) and the other is a twittering trill, which may be confused with the Goldfinch's song.
I reckon I can hear another finch calling in the first example. Any ideas what it is?
The Goldfinch's song is delightful, just like the bird itself and further reason why they were popular caged birds. The song is delivered either from a perch or in flight and is a lovely "isn't life fantastic" tinkling jingle, but also includes various other trills and less pleasing nasal noises. They sing mostly from an elevated perch often with the wings usually slightly drooped and performing the so-called "pivoting display" where it turns from side-to-side in a little jig.
I've not discovered a mnemonic or aide memoir for either the Greenfinch or Goldfinch and have simply had to become familiar with the songs over time.