We have many operating systems, apps and hardware projects, trying to push for a free/libre/open source mobile device. This page attempts to compile all these projects, state the project goals, and to give a real world review. With this you will know what to throw your support behind.
Core Android Apps
These apps intend to replace the "core" system apps that any user would expect. These download/update apps and provide APIs for apps to use.
F-Droid - Repository and app containing only FLOSS software. Has a decent selection. Also check out the Privileged Extension if you are root, so you can install and remove apps in the background without having to confirm.
microG - FLOSS reimplementation of the Google Play Services middleware, a must have if you intend to run most Google Play Apps. (hard to install, you should use a pre-configured custom rom instead).
Yalp Store - Grabs APKs directly from Google Play and facilitates updates.
The Android Open Source Project is licenced under Apache 2.0. However most versions ship with the proprietary Google Play Services ingrained into the OS. Which harms the user.
LineageOS - A fork of Android with it's own set of default apps and extra features. G-Apps are optional and should be avoided if can.
LineageOS with microG - This fork of LineageOS is pre-configured with microG and F-Droid with the privilege extension.
eelo (beta) - New project from a veteran open-source developer/supporter, huge emphasis on privacy and control and attractiveness. It comes with microG, new default apps, account manager, new search engine and online "cloud" services.
Replicant - A fully FLOSS Android OS, that runs absolutely no proprietary firmware/drivers/microcode. Uses F-Droid as it's app store and is approved by the FSF. Only runs on (very) old phones.
Available on F-Droid.
Only available from the publishers website.
Must-have apps, with direct links to either F-Droid or the APK from the official website.
F | Fennec F-Droid - F-Droid fork of Firefox, removes proprietary bits found in Mozilla's builds.
W | ScummVM - Allows you to play old point and click games, plays well on Android.
W | RetroArch - A front-end for emulators, game engines and more. This allows compatible engines (known as cores) to intergrated into one place, and media
F | J2ME Loader - J2ME Emulator. This will play some old Motorola games.
GNU/Linux-based operating systems
These operating systems aren't based on Android, but are instead purpose-built GNU/Linux operating systems.
Ubuntu Touch - Ubuntu ported to mobile, interesting, pleasing and gesture-based UI. Has its own app ecosystem. Works well on the (older but good) "promoted devices". Other devices probably won't work well if at all.
postmarketOS - Touch-optimized, pre-configured Alpine Linux with own packages, designed for mobile devices with the goal of keeping old phones functional and sustainable. Even if a device is supported, it may only be partial. You should check each device individually.
Sailfish - A GNU/Linux OS for mobile, has it's own ecosystem and Android support. However some components (such as the UI) are not FLOSS.
Plasma Mobile - The KDE Plasma desktop environment ported to mobile GNU/Linux!
Anbox - Compatibility layer that allows Android apps to run on any GNU/Linux system.
Shashlik - Provides a way to run Android apps under a GNU/Linux desktop as easily and simply as possible.
Ubuntu Touch Apps
uNav - OpenStreetMap-powered GPS/maps app with offline map function.
Although a fully free SoC is unlikely, There are hardware projects which intend to make everything as free/open as they can. This usually means it will either let you freely install what ever OS, or that it allows hacking (they may even release schematics), or that it runs with a fully free operating system.
ZeroPhone - A work-in-progress open source, Linux-powered, $50 smartphone. Based on the Raspberry Pi Zero, ESP8266 and Arduino.
Librem 5 - A work-in-progress premium smartphone that is intended to respect security, privacy and freedom.
Pine64 Phone - Pine64 sell single-board computers. They are now working on a Linux-powered smartphone planned to run Plasma Mobile. They are also selling low-cost laptops.
Necunos Mobile - A privacy centric open source phone. To-Be-Announced.
Neo900 - An in-dev successor to the N900 phone with modern hardware (including LTE). Can run on a fully FLOSS OS, and allows you to install which ever OS you want.
Halium - A project to standardise all the various mobile GNU/Linux efforts, since much of the low-level base is fragmented across each OS. The project aims to standardise the Linux kernel build and the Android HAL (so non-Android OS's can use Android drivers).
NanoDroid - Flashable package containing microG, F-Droid and other FLOSS apps.