How to escape single quotes within single quoted strings

If you really want to use single quotes in the outermost layer, remember that you can glue both kinds of quotation. Example:

 alias rxvt='urxvt -fg '"'"'#111111'"'"' -bg '"'"'#111111'"'"
 #                     ^^^^^       ^^^^^     ^^^^^       ^^^^
 #                     12345       12345     12345       1234

Explanation of how '"'"' is interpreted as just ':

1. ' End first quotation which uses single quotes.
2. " Start second quotation, using double-quotes.
3. ' Quoted character.
4. " End second quotation, using double-quotes.
5. ' Start third quotation, using single quotes.

If you do not place any whitespaces between (1) and (2), or between (4) and (5), the shell will interpret that string as a one long word.

Since Bash 2.04 syntax $'string' allows a limit set of escapes.

Since Bash 4.4, $'string' also allows the full set of C-style escapes, making the behavior differ slightly in $'string' in previous versions. (Previously the $('string') form could be used.)

Simple example in Bash 2.04 and newer:

  $> echo $'aa\'bb'
  aa'bb

  $> alias myvar=$'aa\'bb'
  $> alias myvar
  alias myvar='aa'\''bb'

In your case:

$> alias rxvt=$'urxvt -fg \'#111111\' -bg \'#111111\''
$> alias rxvt
alias rxvt='urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''

Common escaping sequences works as expected:

\'     single quote
\"     double quote
\\     backslash
\n     new line
\t     horizontal tab
\r     carriage return

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Below is copy+pasted related documentation from man bash (version 4.4):

Words of the form $'string' are treated specially. The word expands to string, with backslash-escaped characters replaced as specified by the ANSI C standard. Backslash escape sequences, if present, are decoded as follows:

    \a     alert (bell)
    \b     backspace
    \e
    \E     an escape character
    \f     form feed
    \n     new line
    \r     carriage return
    \t     horizontal tab
    \v     vertical tab
    \\     backslash
    \'     single quote
    \"     double quote
    \?     question mark
    \nnn   the eight-bit character whose value is the octal 
           value nnn (one to three digits)
    \xHH   the eight-bit character whose value is the hexadecimal
           value HH (one or two hex digits)
    \uHHHH the Unicode (ISO/IEC 10646) character whose value is 
           the hexadecimal value HHHH (one to four hex digits)
    \UHHHHHHHH the Unicode (ISO/IEC 10646) character whose value 
               is the hexadecimal value HHHHHHHH (one to eight 
               hex digits)
    \cx    a control-x character

The expanded result is single-quoted, as if the dollar sign had not been present.

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See Quotes and escaping: ANSI C like strings on bash-hackers.org wiki for more details. Also note that "Bash Changes" file (overview here) mentions a lot for changes and bug fixes related to the $'string' quoting mechanism.

According to unix.stackexchange.com How to use a special character as a normal one? it should work (with some variations) in bash, zsh, mksh, ksh93 and FreeBSD and busybox sh.

I always just replace each embedded single quote with the sequence: '\'' (that is: quote backslash quote quote) which closes the string, appends an escaped single quote and reopens the string.

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I often whip up a "quotify" function in my Perl scripts to do this for me. The steps would be:

s/'/'\\''/g    # Handle each embedded quote
$_ = qq['$_']; # Surround result with single quotes.

This pretty much takes care of all cases.

Life gets more fun when you introduce eval into your shell-scripts. You essentially have to re-quotify everything again!

For example, create a Perl script called quotify containing the above statements:

#!/usr/bin/perl -pl
s/'/'\\''/g;
$_ = qq['$_'];

then use it to generate a correctly-quoted string:

$ quotify
urxvt -fg '#111111' -bg '#111111'

result:

'urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''

which can then be copy/pasted into the alias command:

alias rxvt='urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''

(If you need to insert the command into an eval, run the quotify again:

 $ quotify
 alias rxvt='urxvt -fg '\''#111111'\'' -bg '\''#111111'\'''

result:

'alias rxvt='\''urxvt -fg '\''\'\'''\''#111111'\''\'\'''\'' -bg '\''\'\'''\''#111111'\''\'\'''\'''\'''

which can be copy/pasted into an eval:

eval 'alias rxvt='\''urxvt -fg '\''\'\'''\''#111111'\''\'\'''\'' -bg '\''\'\'''\''#111111'\''\'\'''\'''\'''