I am new here, but have been playing with various Raspberry Pi models for many years. I decided to join the community to share some tricks and knowledge.
I created this video which demonstrates very good sound quality from the PWM output of the Pi Zero and explains how to do this very cheaply yourself - including the schematic (and explanation of what each part does) and configuration files. In the same spirit as the Raspberry Pi Foundation, this video is not monetised by me, so I make no money from you watching this.
I need to clarify that a music claimant has monetised it, not me

(To the moderators: If posting video URLs is not allowed, I apologise)
How to: Get decent analog audio from your Raspberry Pi Zero: https://www.youtube.com/watch?v=ENUb4leJ_rM
The first 10 minutes is an introduction of sorts, then I demonstrate the audio quality until about 18:30, at which point I explain how the Pi generates audio in general, and how to replicate my work. After that, I just play more sound through it for fun

Please take a look, and if you have any questions I am happy to answer them here. The microphone on the camera does NOT do this justice.
This will also improve the audio of the Pi 1 as well mostly due to the output capacitor size and keeping the signals away from noisy parts of the board, so if you still have some of those knocking about and are annoyed with the sound from the built in jack, this might help you too.
For the Pi 2 and Pi 3, this probably will not improve much upon the built in analogue sound jack much (well it might; I have not tried it); on the Pi 2 and onwards the analogue jack is already quite good if you enable sigma delta modulation BUT PLEASE FEED IT TO AN AMPLIFIER - treat it as a line output jack - please!
To be honest, raspi-config should not call it a "headphone" jack - it should explicitly say "3.5mm line-out".