First of all it makes no sense to install versions of Ubuntu that are no longer supported with updates. That is for the reasons 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will never use software newer than is included on the respective install media, only then it might be prudent to use outdated versions.
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 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
nomodset option might help.
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. 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. The Lists of Ubuntu certified hardware might also be of use to you.
Supported versions of Ubuntu
10.04 Server (Lucid Lynx)
The minimum memory requirement for Ubuntu 10.04 LTS is 256 MB of memory.
via Ubuntu Wiki
12.04 Server and Desktop LTS (Precise Pangolin)
700 MHz processor (Intel Celeron or better) - 300 Mhz for Server
384 MB - 128 for Server. Note that some of your system's memory may be unavailable due to being used by the graphics card.
5 GB of hard drive space (1 GB for Server)
VGA capable of 1024x768 screen resolution for Desktop version
12.10 Server and Desktop (Quantal Quetzal)
768 MB of memory and 5 GB of disk space for Ubuntu Desktop [...]. I If your computer has only the minimum amount of memory, the installation process will take longer than normal or even appear frozen for some time.
Supported versions of Lubuntu
Supported versions of Xubuntu
See Xubuntu help page.
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.