I want to replace all single quotes in a string with two single quotes using sed. But when the string contains the & character, the sed command is not repla         
        
It's easier to answer if you post your code, but I'm guessing you're not escaping the ampersand.  Change & to \& if you want a literal ampersand.
See section 3.1.2 of The sed FAQ for a more detailed explantion, if you're curious.
It's working for me
bash>echo "'This is a string with 'single quote' & '&'" | sed "s/'/''/g"
''This is a string with ''single quote'' & ''&''
                                                                        You don't need to escape anything in the input:
$ echo "123 ' foo & b'ar" | sed "s/'/''/g"
123 '' foo & b''ar
However, in the 'replacement' part of the s command & has a special meaning: it means 'match'. That's why the above command can be re-written as:
$ echo "123 ' foo & b'ar" | sed "s/'/&&/g"
123 '' foo & b''ar
Escape it with a \ like everything else that needs to be escaped, if needed:
$ echo "123 ' foo & b'ar" | sed "s/'/'\&'/g"
123 '&' foo & b'&'ar