Case-insensitive search

Yeah, use .match, rather than .search. The result from the .match call will return the actual string that was matched itself, but it can still be used as a boolean value.

var string = "Stackoverflow is the BEST";
var result = string.match(/best/i);
// result == 'BEST';

if (result){
    alert('Matched');
}

Using a regular expression like that is probably the tidiest and most obvious way to do that in JavaScript, but bear in mind it is a regular expression, and thus can contain regex metacharacters. If you want to take the string from elsewhere (eg, user input), or if you want to avoid having to escape a lot of metacharacters, then you're probably best using indexOf like this:

matchString = 'best';
// If the match string is coming from user input you could do
// matchString = userInput.toLowerCase() here.

if (string.toLowerCase().indexOf(matchString) != -1){
    alert('Matched');
}

Suppose we want to find the string variable needle in the string variable haystack. There are three gotchas:

1. Internationalized applications should avoid string.toUpperCase and string.toLowerCase. Use a regular expression which ignores case instead. For example, var needleRegExp = new RegExp(needle, "i"); followed by needleRegExp.test(haystack).
2. In general, you might not know the value of needle. Be careful that needle does not contain any regular expression special characters. Escape these using needle.replace(/[-[\]{}()*+?.,\\^$|#\s]/g, "\\$&");.
3. In other cases, if you want to precisely match needle and haystack, just ignoring case, make sure to add "^" at the start and "$" at the end of your regular expression constructor.

Taking points (1) and (2) into consideration, an example would be:

var haystack = "A. BAIL. Of. Hay.";
var needle = "bail.";
var needleRegExp = new RegExp(needle.replace(/[-[\]{}()*+?.,\\^$|#\s]/g, "\\$&"), "i");
var result = needleRegExp.test(haystack);
alert(result);

If you're just searching for a string rather than a more complicated regular expression, you can use indexOf() - but remember to lowercase both strings first because indexOf() is case sensitive:

var string="Stackoverflow is the BEST"; 
var searchstring="best";

// lowercase both strings
var lcString=string.toLowerCase();
var lcSearchString=searchstring.toLowerCase();

var result = lcString.indexOf(lcSearchString)>=0;
alert(result);

Or in a single line:

var result = string.toLowerCase().indexOf(searchstring.toLowerCase())>=0;

Replace

var result= string.search(/searchstring/i);

with

var result= string.search(new RegExp(searchstring, "i"));