Which characters are illegal within a branch name?

The accepted answer and the man page already explain which rules apply to Git branch names.

In the Git source code, the refname_disposition array is used to determine how to handle various characters in refnames. The indexes in the array correspond to ASCII codes and the values indicate how the ASCII characters are handled.

/*
 * How to handle various characters in refnames:
 * 0: An acceptable character for refs
 * 1: End-of-component
 * 2: ., look for a preceding . to reject .. in refs
 * 3: {, look for a preceding @ to reject @{ in refs
 * 4: A bad character: ASCII control characters, and
 *    ":", "?", "[", "\", "^", "~", SP, or TAB
 * 5: *, reject unless REFNAME_REFSPEC_PATTERN is set
 */
static unsigned char refname_disposition[256] = {
    1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
    4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
    4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 1,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 4, 4
};

Since 4 means that the corresponding ASCII character is not allowed inside branch names, there are 39 disallowed characters. The disallowed characters are ASCII control characters (ASCII codes < 32), the printable characters : ? [ \ ^ ~ and the whitespace/tab character.

3 more characters require some conditions to be met (see doc comment):

• .: Two subsequent dots are forbidden.
• {: The substring @{ is forbidden.
• *: Reject unless REFNAME_REFSPEC_PATTERN is set.

The null byte terminates the branch name and / creates a new directory hierarchy for the branch. Therefore, branch names cannot end with a slash. For example git checkout -b 'a/b/c' will create the corresponding directory structure under .git/refs/heads

Note that UTF-8 characters can be used in branch names:

$ git checkout -b $'\xCE\xA9'
Switched to a new branch 'Ω'

As an addition, care must be taken if you consider to use the dollar sign $ character. git branch pew$ign will create pew. In order to create a branch that has $ within it the whole name should be wrapped in quotes: git branch 'pewSign'. Ideally you should avoid to use the symbol whatsoever.

It doesn't like > or ==> or ->

To complete the main answer by Manoj Govindan :

• @ is a valid branch name (on git version git version 2.24.3 (Apple Git-128))
• HEAD is not a valid branch name (by captain obvious!)
• length is limited (by the OS, on Mac OS 10.15.7, 250 characters is the maximum, either ascii or not-ascii)