TL/DR

Sed expressions containing $BASH variables need to be double (")-quoted for the variable to be interpreted correctly.

If you also double-quote your $BASH variable (recommended practice)

… then you can escape the variable double quotes as shown:

sed -i "s/foo/bar ""$VARIABLE""/g" <file>

I.e., replace the $VARIABLE-associated " with "".

(Simply -escaping "$VAR" as \"$VAR\" results in a "-quoted output string.)


Examples

$ VAR='apples and bananas'
$ echo $VAR
apples and bananas

$ echo "$VAR"
apples and bananas

$ printf 'I like %s!\n' $VAR
I like apples!
I like and!
I like bananas!

$ printf 'I like %s!\n' "$VAR"
I like apples and bananas!

Here, $VAR is "-quoted before piping to sed (sed is either '- or "-quoted):

$ printf 'I like %s!\n' "$VAR" | sed 's/$VAR/cherries/g'
I like apples and bananas!

$ printf 'I like %s!\n' "$VAR" | sed 's/"$VAR"/cherries/g'
I like apples and bananas!

$ printf 'I like %s!\n' "$VAR" | sed 's/$VAR/cherries/g'
I like apples and bananas!

$ printf 'I like %s!\n' "$VAR" | sed 's/""$VAR""/cherries/g'
I like apples and bananas!

$ printf 'I like %s!\n' "$VAR" | sed "s/$VAR/cherries/g"
I like cherries!

$ printf 'I like %s!\n' "$VAR" | sed "s/""$VAR""/cherries/g"
I like cherries!

Compare that to:

$ printf 'I like %s!\n' $VAR | sed "s/$VAR/cherries/g"
I like apples!
I like and!
I like bananas!


$ printf 'I like %s!\n' $VAR | sed "s/""$VAR""/cherries/g"
I like apples!
I like and!
I like bananas!

… and so on …


Conclusion  

My recommendation, as standard practice, is to

  • "-quote BASH variables ("$VAR")
  • "-quote, again, those variables (""$VAR"") if they are used inside a sed expression (which itself must be "-quoted, not '-quoted)
$ VAR='apples and bananas'

$ echo "$VAR"
apples and bananas

$ printf 'I like %s!\n' "$VAR" | sed "s/""$VAR""/cherries/g"
I like cherries!


References

Cross-posted to StackOverflow: How Do I Escape Double and Single Quotes in sed?

See also: