why does git reflog HEAD@{0} return ambiguous argument

Unix-like systems have command interpreters called shells. Popular shells include bash and zsh, but there are many more. Each shell has its own special quirks.

It's very common for shells to support file name globbing: e.g., echo * has * expand to all file names, so that echo does not have to do that. I'd be surprised to find a shell that didn't do this (although anything is possible!). This means that if you want to pass a literal * to any command including Git—Git will do its own glob-style expansion, which won't match the shell's in various cases—you need to quote the asterisk. Otherwise the shell says: Aha, an asterisk, I must replace this with something else. So echo '*' makes echo print the literal character *.

The curly brace character gets this kind of treatment sometimes in some shells. For instance, in csh or tcsh, { participates in brace-expansion:

tcsh$ echo {foo,bar}{baz,quux} foobaz fooquux barbaz barquux 

Here, HEAD@{1} triggers brace-expansion, which dumps HEAD@1 into the final Git command. That's not quite what you're seeing—you got HEAD@, with the {1} part stripped—so clearly you are not using either csh or tcsh.

The bash shell does the same kind of trick, but is a bit cleverer:

bash$ echo {foo,bar}{baz,quux} foobaz fooquux barbaz barquux bash$ echo HEAD@{1} HEAD@{1} 

Because there is no comma inside the {1},¹ bash left the whole thing alone and hence passed the right thing to echo.

It's clear that whatever shell you are using, it did something with the {1} part. What, and why, will depend on whatever shell you are using. Most shells obey simple quoting rules where single quotes quote everything, though, so 'HEAD@{1}' should work. I tend to use HEAD@\{1} or HEAD@'{1}', which also work, when I'm in this kind of situation. Consult the documentation for your particular shell for more details.

¹Bash's brace-expansion also treats double-dot specially when combined with numeric values or single characters; csh/tcsh's does not.