When logging in through SSH:

System information disabled due to load higher than 1.0

GUI version : Space Radar

Install Space Radar on a client machine. On the server, generate a txt sizes file with du -ak

$ du -ak /var/log /usr | gzip -c > /tmp/sizes.txt.gz

See Space Radar repo

CLI version

Which partition is full?

$ df
Filesystem 1K-blocks Used Available Use% Mounted on
udev 991316 0 991316 0% /dev
tmpfs 199908 20936 178972 11% /run
/dev/vda1 10287416 9833040 4196 100% /
tmpfs 999528 0 999528 0% /dev/shm
tmpfs 5120 24 5096 1% /run/lock
tmpfs 999528 0 999528 0% /sys/fs/cgroup
cgmfs 100 0 100 0% /run/cgmanager/fs
tmpfs 199908 0 199908 0% /run/user/1000

Which directory is full?

$ du -h --max-depth=1 / | sort -hr
9,4G /
5,2G /var
2,1G /usr
1,4G /opt
560M /root
158M /lib
57M /boot
49M /home
21M /run
13M /etc
13M /bin
9,9M /sbin
3,9M /lib32
3,1M /tmp
$ du -h --max-depth=2 /home | sort -hr

If you still have some Mb left, ncdu (ncurse disk usage) might help:

$ ncdu
$ ncdu ~
$ sudo ncdu /
# inside ncdu, quit with q
# and go up folder with <backspace>

Common temporary files safe to remove

Delete rotated logs

$ ll /var/log/*.{gz,bz2} # check
$ rm /var/log/*.{gz,bz2} # remove