My Mac mini 2012 with Catalina crashes at boot

Start with Use Apple Diagnostics to test your Mac

1 - Restart in Safe Mode. This will perform a Disk Repair, clear cache files and only load Apple Software, extensions and fonts. The boot up will be slow and can take some time - Normal.

2 - Does the issue present in this mode ?

3 - Sometimes a Safe Boot followed by a Normal Boot will just put things right.

How was the computer configured to Boot ?

Was the computer set to Boot to an External Drive in System Preferences >> Startup Disk ?

Was the User attempting to Erase the Drive and reinstall the Operating System as new ?

How to reinstall macOS

Follow these guidelines during installation:

• If the installer asks to unlock your disk, enter the password you use to log in to your Mac.
• If the installer doesn't see your disk, or it says that it can't install on your computer or volume, you might need to erase your disk first.
• If the installer offers you the choice between installing on Macintosh HD or Macintosh HD - Data, choose Macintosh HD.

This all begs to ask what happened that caused your mini to start crashing at boot up. This is the only thing you didn't provide us in your initial post. For example, was it running fine, and then, started crashing at boot up? Did you update/upgrade macOS or make any other changes? Did you add a new peripheral? Something else?

If your Mac's UEFI firmware is intact and the associated disk image for recovery or diagnostics is not corrupted, your Mac should be able to boot up in either mode. Otherwise, it will attempt to do the same with the Internet recovery or diagnostics modes. Its the UEFI that checks which key combination you use at boot up.

At this point, I would suggest you run the Apple Hardware Test (AHT) local diagnostic. This should point out any significant hardware issues with your Mac. With the mini powered-down, to run the AHT. hold down the <D> key, and then press the power button. If it fails to load the local diagnostics, it should immediately attempt to go to Internet diagnostics. This may take a bit of time to start. If either are successful, it should start running the diagnostics. Allow it to complete, and note the results. Please post them here.

Ref: How to use Apple Hardware Test on your Mac - Apple Support

Should everything check out good, we can next move on to recovery mode to reinstall macOS.

Either you don't have a good network/Internet connection acceptable for this procedure, or a hardware problem is interfering (most likely a bad hard drive as the others have already mentioned).

You can create & use a bootable Knoppix USB stick to check the health of the hard drive. It also allows you to see if the laptop will boot and stay powered on. Download the Knoppix DVD .iso file with "EN" in the name for "ENglish". Use this .iso file as a source Etcher (Mac/Windows/Linux) which will be used to make the bootable Knoppix USB stick. Option Boot the Knoppix USB stick and select the orange icon labeled "EFI". While booting Knoppix, the Mac may appear frozen on the Apple boot picker menu so make sure to give Knoppix lots of time to finish booting.

Once Knoppix boots to the desktop, click the "Start" menu icon on the Taskbar and navigate the menus to "System Tools --> GSmartControl". Within GSmartControl double-click on the laptop's internal drive icon to access the drive's health report. Post the complete report here.

If Knoppix only boots the command prompt, then I can provide other instructions to access the drive's health report.

You should always have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data.

samnegri wrote:

I booted into safe mode and was offered two options, either boot using OS X Base system or launch from my external drive.

The "OS X Base System" is the Mac's recovery partition. It contains the minimal Booter files, kernel, and the BaseSystem.dmg disk image of a basic copy of macOS. It typically is only mounted when your Mac is in recovery mode and is used to install macOS without a CD.

At some point, your Mac's operating system was either erased or corrupted. @P. Phillips has already provided you with an excellent set of tips on how to get this resolved.

Power-down your Mac. This time hold down the <option><D> key combination, and then, apply power. This should start Internet diagnostics ... not Internet recovery. Again, it may take a while for it to start. Did it? If not, then (again) it's may be a failed hard drive.

Must ask very pointedly again >>

Q -" I booted into safe mode and was offered two options, either boot using OS X Base system or launch from my external drive. "

A - Was the computer set to Boot to an External Drive in System Preferences >> Startup Disk ?

That may also indicate that the drive is either failing or has already failed ... especially if it is the original drive that came with this Mac.