How to activate root using the Single-User Mode in Catalina
With the slightly different volume scheme in Catalina - compared to pre-Cataline systems - you have to mount /System/Volumes/Data to prevent the read-only mount point /:

/sbin/mount -uw /System/Volumes/Data


Then the usual commands:

launchctl load /System/Library/LaunchDaemons/com.apple.opendirectoryd.plist
passwd root


and enter the password.

On T2-chip Macs you have to boot to Recovery Mode and use dscl to enable root:

dscl -f /Volumes/Macintosh\ HD\ -\ Data/private/var/db/dslocal/nodes/Default localhost -passwd /Local/Default/Users/root


If your main volume has another name than Macintosh HD, replace Macintosh\ HD with this name (e.g SystemCat):

dscl -f /Volumes/SystemCat\ -\ Data/private/var/db/dslocal/nodes/Default localhost -passwd /Local/Default/Users/root