Trying to upload BigSur. It progresses about 3/4 and then stops and won't continue??

Try the following:

#1 - boot into Safe Mode according to Start up your Mac in safe mode - Apple Support.

NOTE: Safe Mode boot can take up to 3-5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

#2 - go to the System/Software Update preference pane and begin the download.

#3 - run the installer from the Safe Mode.

I've had better luck from the Safe Mode because I evidently have some 3rd party software that interferes with the installation.

What is “Other” storage on a Mac, and how can I clean it out?

Free up storage space on your Mac

OmniDiskSweeper safe to use

See used and available storage space on your Mac

• Other: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.

As @Viking OSX stated earlier the Empty Space Requirements and suggest at least 50 GB.

Sorry if this sounds harsh - but thinking there is enough Empty Space is not the same as knowing there is enough Empty Space.

The worst thing that could happen, but does not mean will happen - the 12.24 GB Full Installer Downloads and attempts to install - and presto - it does but the computer can not boot because of the missing Empty Space.

Double check if there is an Applicaiton in Applications folder called " Install Big Sur " If it exists, drag and drop to Trash and Empty Trash. At the sometime, check in Download folder in the Dock - If there, same thing Trash it.

1 - Connect to Router via Ethernet connection and restart the computer in Safe Mode and then open the Apple Apps Store and get the Full Installer of Big Sur. It will be the 11.6 version and once downloaded run the installer from the Applications folder.

Or

3 - This method may seem unnecessary but sometimes the Router Restart clears Network issues from ISP to Apple Servers

Shutdown Router. Shutdown computer and disconnect all External Drives. Restart Router. Connect to Router via Ethernet Cable. If via Wifi - Disconnect all other Devices using the Wifi - want the Maximum Bandwidth only for this computer. Restart computer in  Safe Mode. It does a Repair Disk, Clear System Cache files and only load Apple Software, extensions and Fonts. Will load slowly - Normal. Once Logged in - open Apple Apps Store and  Get the Full Version of Big Sur. It will take some time and several pauses will occur - normal. Countdown clock may say 15 minutes but in Real time may be 30 minutes. Same goes for clock at 10 minutes could be 20 minutes in Real time. Eventually , Full Installer will download and a New Application in Applications Folder will present called Install Big Sur. Stop there and perform a Time Machine Backup. If desired - have an External Drive - Copy & Paste the Install Big Sur Application to external drive for future usage - making a Bootable Installer. Now run the Installer.

Control (right) - click on your hard drive's icon on the Desktop and select Get Info from the contextual menu. In the Info window will the the used and free space.

The files that you have control over are located in the Documents, Downloads, Pictures, Music and Movies folders.  You can use either of these two free apps, GrandPerspective  or OmniDiscSweeper, to find the largest files on your drive so you can determine if they can be deleted or moved to an external HD for storage.  

Note: you can empty the Downloads folder after the apps and/or updates that were downloaded have been installed or applied.  Many users have found a couple of Gigabytes of files in their Downloads folder which are no longer needed. 

You will need at least 35 GB of available storage to download and install Big Sur. It is a monster and takes a good while to install with some apparent no-activity delays along the way. Patience is a virtue here.

If this upgrade is for Big Sur 11.6 the error message can be misleading. There are factors:

1 - Like poor Wifi connection

2 - Slow wifi connection

3 - Too many other people downloading at the same time and Apple Severs are overloaded

4 - Software on the computer interfering with the download process including AntiVirus Software

5 - Running behind a VPN which may increase Packet losses

6 - If via wifi too many Other devices using the wifi at the same time reducing the Bandwidth to your computer. 

7 - Best Results are via Ethernet Cable to Router

It should be as the installer is about 12GB in size. However, after installation check the free space and try to maintain at least 65 GB free or more. It will help in improving system and application performance.

Only take it to an Apple Genius bar or an Authorized Apple Service Provider. Don';t rely on some local computer shot.