How do I find out which version and derivative of Ubuntu is right for my hardware in terms of minimal system requirements?

Preliminary notes

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First of all it makes no sense to install versions of Ubuntu that are no longer supported with updates. The reasoning is discussed at the bottom of this answer.

This answer will concentrate on currently supported versions of Ubuntu and its official derivatives.

If your hardware never connects to the internet and if you will never use software newer than is included on the respective install media, only then might it be prudent to use outdated versions. But who can be sure of that for all eternity?

You don't have to install Ubuntu to see if it works on your hardware. It is always a good idea to boot from live DVD/USB and see if the system runs okay (although slowly) on the given hardware.

Even if it seems not to work, you might be just one boot option away from a working system. See My computer boots to a black screen, what options do I have to fix it? For example the nomodeset option might help.

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Currently supported versions and their minimal requirements

The community wiki usually offers an up-to-date list of currently supported versions. Minimal system requirements for Ubuntu can be found there in the tech specs. The Lists of Ubuntu certified hardware might also be of use to you.

Information on the derivatives is scarce but their requirements are less than what is listed here. Generally 32-bit versions take up less memory and tend to be faster on older systems, than their 64-bit counterparts. However, allot of software nowadays assumes 64-bit architecture. If you are unsure, install the 64-bit versions.

Supported versions of Ubuntu

• 20.04 and later Desktop recommended system requirements

• 2GHz dual core processor or better

• 4GB RAM

• 25GB of disk space

• VGA capable of 1024x768 screen resolution

• Either a DVD drive or a USB port for the installation media

• Internet access is helpful

• 16.04 Desktop recommended system requirements

• 2GHz dual core processor or better

• 2GB RAM

• 25GB of disk space

• Graphics processor and display capable of at least 1024x768

• Either a DVD drive or a USB port for the installation media

• Internet access is helpful

• 16.04 Desktop minimal

In order to run Unity the system needs a more capable graphics adapter.

• Intel Pentium 4 processor or better

• 1GB RAM

• Graphics processor and display capable of at least 1024x768

• 10 GB of disk space

• 17.10 and later Desktop recommended system requirements

• 64-bit 2GHz dual core processor or better

• 4GB RAM

• 25GB of disk space

• Graphics processor and display capable of at least 1024x768

• Either a DVD drive or a USB port for the installation media

• Internet access is helpful

• 17.10 and later Desktop minimal

• 64-bit processor for iso, Intel Pentium 4 processor or better for installation using the Ubuntu minimal CD. The Ubuntu minimal CD (for Ubuntu 18.04 and earlier) supports installation of Ubuntu on computers that have a 32-bit processor.

• 1GB RAM

• Graphics processor and display capable of at least 1024x768

• 10 GB of disk space

Updates will be provided for Ubuntu 18.04 for 5 years until April, 2023. Updates will be provided for Ubuntu 19.10 for 3 months until July, 2020.

• 16.04 and 18.04 Server installation standard

  • Ubuntu Server supports 3 major architectures: Intel x86, AMD64 and ARM.
  • 1 GHz processor
  • 512 MB of system memory (RAM)
  • 1 GB of disk space (base system)
  • 1.75 GB of disk space (all tasks installed)
  • Graphics card and monitor capable of 640x480

• 16.04-20.04 Server installation minimal

  • Ubuntu Server supports 3 major architectures: Intel x86, AMD64 and ARM.
  • 300 MHz processor
  • 256 MB of system memory (RAM) 64-bit, 192 MB RAM 32-bit
  • 700 MB of disk space (base system)
  • 1.4 GB of disk space (all tasks installed)
  • Graphics card and monitor capable of 640x480

Security and maintenance updates will be provided for Ubuntu Server 16.04 until April, 2021 and for Ubuntu Server 20.04 until April, 2025.

Supported versions of Xubuntu

See Xubuntu help page.

• 16.04-18.10

Minimum systems requirements
To install or try Xubuntu within the Desktop/Live CD, you need 512MB of memory, 700 MHz processor and 7.5GB of free space on your hard disk.

Recommended systems requirements
To get a smooth experience when running multiple applications in parallel on the desktop, it is recommended to have at least 1GB of memory. It is recommended to have at least 20GB of free disk space. This allows new application installations as well as saving your personal data on the hard disk in addition to the core system.

Security and maintenance updates will be provided for Xubuntu 16.04 for 3 years until April, 2019 and for Xubuntu 20.04 for 3 years until April, 2023. Security and maintenance updates will be provided for Xubuntu 19.10 for 9 months until July, 2020. 

• 19.04 and later

Minimum systems requirements
To install or try Xubuntu within the Desktop/Live CD, you need 512MB of memory, 64-bit 700 MHz processor and 7.5GB of free space on your hard disk.

Recommended systems requirements
To get a smooth experience when running multiple applications in parallel on the desktop, it is recommended to have at least 1GB of memory. It is recommended to have at least 20GB of free disk space. This allows new application installations as well as saving your personal data on the hard disk in addition to the core system.

Supported versions of Lubuntu

• 16.04-17.10

Processor (CPU)
The minimum specification for CPU is Pentium 4 or Pentium M or AMD K8. Older processors are too slow and AMD K7 has problems with flash video.

Memory (RAM)
For advanced internet services like Google+, YouTube, Google Docs and Facebook, your computer needs at least 1GB RAM. Install Lubuntu 32-bit for computers with <= 3GB RAM.

For local programs like LibreOffice and simple browsing habits, your computer needs at least 512MB RAM.

• 18.04 and 18.04

Processor (CPU)
The minimum specification for CPU is Pentium 4 or Pentium M or AMD K8. Older processors are too slow and AMD K7 has problems with flash video.

Memory (RAM)
For advanced internet services like Google+, YouTube, Google Docs and Facebook, your computer needs at least 2GB RAM. Install Lubuntu 32-bit for computers with <= 3GB RAM.

For local programs like LibreOffice and simple browsing habits, your computer needs at least 1GMB RAM.

• 19.04 and later

Processor (CPU)
64-bit processor

Memory (RAM)
For advanced internet services like Google+, YouTube, Google Docs and Facebook, your computer needs at least 2GB RAM.

For local programs like LibreOffice and simple browsing habits, your computer needs at least 1GMB RAM.

Lubuntu Alternate ISOs are for low-RAM PCs. Computers with less than 700 MB of RAM are considered low-RAM computers. Instructions are here.

Updates will be provided for Lubuntu 20.04 for 3 years until April, 2023. Updates will be provided for Lubuntu 19.10 for 9 months until April, 2020.

Supported versions of Ubuntu Core

Ubuntu Core, with an image size of 260MB, is the smallest Ubuntu release to date. This makes it ideal both for IoT devices and cloud containers. A new release of Ubuntu Core is released every 2 years on even numbered years (2016, 2018, 2020, etc.).

Processor – 600MHz processor (ARMv7 or greater, or x86)
System Memory – 128MB RAM or greater
Storage – 4GB flash / storage for factory reset and system rollback

• Ubuntu Core 16 and 18

Ubuntu Core 16 is based on Ubuntu 16.04, and is supported for 5 years.

Ubuntu Core 18 is based on Ubuntu 18.04, and is supported for 10 years.

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^{RAM Usage of different flavors of 18.04 under baseline conditions  (Click image to enlarge)}

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Why you shouldn't use versions when their support has ended

• Security risks: Eventually there will be an exploit that compromises security or system integrity of old Ubuntu versions
• Software incompatibilities: Versions that are no longer supported will have increasing problems with this. Due to the lack of updates one will no longer be able to open the most recent LibreOffice documents or compile programs that need more recent libraries. Hardware drivers of recent devices will not be included in older kernels.
• Decreasing availability of repositories: It might become very difficult to download software that does not already ship with the outdated version. Hosting repositories for very old versions cease to be economically viable at one point.

Lubuntu or Xubuntu (Ubuntu with the Xfce desktop environment). Xubuntu is more "user friendly" -- more graphical tools for settings, better looking and better integrated applications and maybe better support (larger community). On the other hand, Lubuntu 14.04 LTS needs less RAM (Lubuntu 14.04 needs 128 MB, Lubuntu 15.04-17.04 needs 512 MB, Lubuntu 18.04 and later needs 1 GB, and Xubuntu 14.04 and later needs 512 MB). Xubuntu and Lubuntu 19.04 and later are available for 64-bit architecture only. For advanced internet services like Google+, YouTube, Google Drive, and Facebook, your computer needs at least 1 GB of RAM. It depends on your skills and preferences, you can try both and then choose. Lubuntu Alternate ISOs are for low-RAM PCs. Computers with less than 700 MB of RAM are considered low-RAM computers. Instructions are here.

• Lubuntu: https://lubuntu.me
• Xubuntu: https://xubuntu.org

^{Lubuntu 19.10 with the LXQt desktop environment  (Click image to enlarge)}

Xubuntu

Xubuntu is designed to run on lightweight machines. I recently installed it on a machine with 768MB of RAM and other specs close to your machine and it runs without an issue.

^{Xubuntu 19.10  (Click image to enlarge)}

Xubuntu is simply Ubuntu bundled with the Xfce Desktop Environment, a desktop environment designed to be fast and lightweight on lower end systems without compromising performance and visual style. Xubuntu 20.04 is a Long Term Support release for which security and maintenance updates will be provided for 3 years until April, 2023.

Xubuntu 16.04 needs at least 512 MB RAM, 700 MHz processor and 6.1 GB of free space on your hard disk (minimum) and 20 GB free space (recommended). Xubuntu 20.04 is a Long Term Support release for which security and maintenance updates will be provided for 3 years until April, 2023.

Xubuntu 18.04-20.04 needs at least 512 MB RAM, 700 MHz processor and 7.5 GB of free space on your hard disk (minimum) and 20 GB free space (recommended). Xubuntu 20.04 is a Long Term Support release for which security and maintenance updates will be provided for 3 years until April, 2023.

Xubuntu 19.04 and later is available for 64-bit architecture only.