How to make git status show only staged files

Use git diff --name-only (with --cached to get the staged files)

The accepted answer won't let you know what kind of changes were there.

Yes, If you are not syntax checker but an ordinary person with a repository full of unstaged files, and you still want to know what will happen to staged files - there is another command:

git status --short | grep '^[MARCD]'

which leads to something like:

M  dir/modified_file
A  dir/new_file
R  dir/renamed -> dir/renamed_to
C  dir/copied_file
D  dir/deleted_file

Obviously, this files were staged, and after git commit:
deleted_file will be deleted,
new_file will be added,
renamed_file will become a renamed_to.

Here is an explanation of short-format output: https://git-scm.com/docs/git-status#_short_format

Inspired by @coffman21's answer I have setup the following alias in my .zshrc

alias gst="git status"
alias gst-staged="git status --short | grep '^\w.'"
alias gst-unstaged="git status  --short | grep '^\W.'"
alias gst-unstaged-tracked="git status  --short | grep '^\s.'"
alias gst-untracked="git status --short | grep '^??'"

or

alias gst="git status"
alias staged="git status --short | grep '^\w.'"
alias unstaged="git status  --short | grep '^\W.'"
alias unstaged-tracked="git status  --short | grep '^\s.'"
alias untracked="git status --short | grep '^??'"

It might be of use to anyone else. So adding it to the stack of answers.

to view staged files with code changes

git diff --staged   

or using --cached which is synonym for --staged

git diff --cached

or to view only file names without code changes

git diff --staged --name-only  

git-diff manual