Single User Mode not starting
Although this does not answer the original question, as it is about an earlier mac model, this is a somewhat common issue for newer macs, and I feel that this answer may be helpful from the people that come here from Google, where this page is one of the top results.

If, when entering single user mode, you encounter the issue shown in this question's GIF (the mac shows the single user mode white text, but never prompts a terminal and just contiunes booting anyway), the answer is that Single User Mode via CMD+S does not work on T2 macs (if your mac's model appears on this list, then it has a T2 chip).
Apple's official guidance (which strangely now leads to a 404 page, but is still available on the Wayback Machine) is to:


Turn on your Mac, then immediately press and hold CMD+R to start up from macOS Recovery.
Select Disk Utility from the Utilities window, then click Continue.
From the Disk Utility sidebar, select the volume that you're using, then choose File &gt; Mount from the menu bar. (If the volume is already mounted, this option is dimmed.) Then enter your administrator password when prompted.
Quit Disk Utility.
Choose Terminal from the Utilities menu in the menu bar.
You can now enter UNIX commands. When done, choose Apple () menu &gt; Restart.

These steps replace the CMD+S method, which doesn't work with the recommended settings on current Mac computers.