Where are the snapshots saved? [duplicate]

The actual data is stored in a hidden directory called .MobileBackups at the root of the volume.

There is a special virtual filesystem type called mtmfs which translates the raw data in these hidden directories into a virtual Time Machine drive which is automatically mounted on /Volumes/MobileBackups

This mechanism of storing local backups existed on HFS format file systems and is no longer how things work on APFS Macs. Snapshots now are built in to the filesystem as opposed to needing to make “shadow copies” of files in a new location, the “shadow copies” are of the whole disk state.

• Now that disablelocal is gone, how to stop local (internal drive) TimeMachine backups?

Under the new file system in High Sierra (Apple File System (APFS)), this it is different. You can see what is considered the "local backups" via command line like this:

sudo tmutil listlocalsnapshots /

Ones that are listed as "(dateless)" will cause all sorts of problems:

com.apple.TimeMachine.2017-12-23-090037 (dataless)

so you can delete any backup by command line like this (deletes the one listed above, note only date used in reference):

sudo tmutil deletelocalsnapshots 2017-12-23-090037

Note that in my latest problem with macOS, the "bird" process was eating up egregious amounts of CPU. It was suggested that dumping backups that were damaged (damaged, who knew?) would help.

They are stored on your hard drive in the folder entitled:

/Volumes/MobileBackups

Note: This answer is now dated and incorrect for current MacOS versions. It was accurate for MacOS Lion, but has changed since that time. See other answers for updated locations.

On Catalina they seem to be in:

/Volumes/com.apple.TimeMachine.localsnapshots