Using git show

To complete your own answer, the syntax is indeed

git show object
git show $REV:$FILE
git show somebranch:from/the/root/myfile.txt
git show HEAD^^^:test/test.py

The command takes the usual style of revision, meaning you can use any of the following:

1. branch name (as suggested by ash)
2. HEAD + x number of ^ characters
3. The SHA1 hash of a given revision
4. The first few (maybe 5) characters of a given SHA1 hash

Tip It’s important to remember that when using “git show“, always specify a path from the root of the repository, not your current directory position.

(Although Mike Morearty mentions that, at least with git 1.7.5.4, you can specify a relative path by putting “./” at the beginning of the path. For example:

git show HEAD^^:./test.py

)

Using git restore

With Git 2.23+ (August 2019), you can also use git restore which replaces the confusing git checkout command

git restore -s <SHA1>     -- afile
git restore -s somebranch -- afile

That would restore on the working tree only the file as present in the “source” (-s) commit SHA1 or branch somebranch.
To restore also the index:

git restore -s <SHA1> -SW -- afile

(-SW: short for --staged --worktree)

---

As noted in the comments by starwarswii

It lets you pipe the contents into a file, which is great if you want to just quickly compare files from a commit.

E.g. you can do:

git show 1234:path/to/file.txt > new.txt 
git show 1234~:path/to/file.txt > old.txt

then compare them.

---

Using low-level git plumbing commands

Before git1.5.x, this was done with some plumbing:

git ls-tree <rev>
show a list of one or more ‘blob’ objects within a commit

git cat-file blob <file-SHA1>
cat a file as it has been committed within a specific revision (similar to svn
cat).
use git ls-tree to retrieve the value of a given file-sha1

git cat-file -p $(git-ls-tree $REV $file | cut -d " " -f 3 | cut -f 1)::

git-ls-tree lists the object ID for $file in revision $REV, this is cut out of the output and used as an argument to git-cat-file, which should really be called git-cat-object, and simply dumps that object to stdout.

---

Note: since Git 2.11 (Q4 2016), you can apply a content filter to the git cat-file output.

See
commit 3214594,
commit 7bcf341 (09 Sep 2016),
commit 7bcf341 (09 Sep 2016), and
commit b9e62f6,
commit 16dcc29 (24 Aug 2016) by Johannes Schindelin (dscho).
^{(Merged by Junio C Hamano — gitster — in commit 7889ed2, 21 Sep 2016)}

git config diff.txt.textconv "tr A-Za-z N-ZA-Mn-za-m <"
git cat-file --textconv --batch

Note: “git cat-file --textconv” started segfaulting recently (2017), which has been corrected in Git 2.15 (Q4 2017)

See commit cc0ea7c (21 Sep 2017) by Jeff King (peff).
^{(Merged by Junio C Hamano — gitster — in commit bfbc2fc, 28 Sep 2017)}