Copying text from a terminal
Most mature terminal emulators permit users to copy or save their contents.
General approach
    
In graphical terminal emulators, contents are typically selectable by mouse, and can then be copied using the context menu, Edit menu or a key combination such as Ctrl+Shift+C.
Xorg
    
Some emulators do not support the CLIPBOARD selection natively, and copy data to the PRIMARY selection. For them xclip may be used:
$ xclip -o | xclip -selection clipboard -i
The above command reads data from the PRIMARY selection and writes it to CLIPBOARD selection.
Other clipboard managers such as autocutsel provide automatic synchronization between selection buffers.
Intercepting commands output
    
Use tee to intercept the output of a command.
$ command 2>&1 | tee output-file
After the command is executed, output-file will contain its output, while having displayed the output at the same time.
Accessing Linux terminal backlog
    
The backlog of a native terminal named /dev/ttyN may be accessed via /dev/vcsN.
Hence, if one is working in /dev/tty1, the following snippet will let store the backlog in a file output-file:
# cat /dev/vcs1 >output-file
Comparison of common emulators
    
Unless the "Key combination" column states otherwise, the key combination is Ctrl+Shift+c.
| Emulator | Select to PRIMARY | CLIPBOARD | ||||
|---|---|---|---|---|---|---|
| Key combination | Context menu | Window menu | Select | |||
| Alacritty | No | No | No | |||
| No | No | No | No | |||
| No | No | No | No | |||
| No | No | |||||
| Guake | No | No | ||||
| Konsole | Optional | |||||
| lilyterm-gitAUR | No | No | ||||
| No | ||||||
| No | ||||||
| No | No | |||||
| No | No | |||||
| PuTTY | No | No | No | No | ||
| No | ||||||
| No | ||||||
| No | No | No | No | |||
| sakuraAUR | No | |||||
| st | No | No | No | |||
| Terminator | No | No | ||||
| terminology | No | No | ||||
| Termite | No | No | No | |||
| Tilda | No | No | ||||
| urxvt | No | No | Optional | |||
| No | ||||||
| xterm | Optional | No | No | |||
| Yakuake | No | Optional | ||||
Special cases
    
    putty
    
The xclip approach works for putty: one just has to remember that the xclip invocation should be done on the local computer (in another terminal), not on the remote machine to which putty is connected.
urxvt
    
Selecting text to CLIPBOARD requires the perl extension. See Rxvt-unicode#Cut and paste for details.
xterm
    
Access to the CLIPBOARD selection in xterm requires additional steps.
mlterm
    
In addition to Ctrl+Shift+c, you can use  if you do not want to kill processes accidentally.