How can I stay tab-free in Geany on Ubuntu?

You are looking for 'soft tabs'

http://www.geany.org/manual/dev/index.html#editor-indentation-preferences

Edit > Preferences > Editor > Indentation > Type > Spaces

After changing the setting, close and reopen the file.

To apply the tab settings to every document in the current session, also click:
Project->Apply Default.

DTing is right but on thing to remember: You have to close all files (tabs) in Geany and open them up again to have the new settings work in each file. It took me days to figure out that I not only have to close geany and open it up again but that I also have to re-open each file seperately.

I know I'm a bit late on this, but hopefully this answer might help someone else. With Geany closed, edit ~/.config/geany/geany.conf with some other editor, changing the lines:

use_tab_to_indent=true
indent_type=1

to:

use_tab_to_indent=false
indent_type=0

and the settings will "take".