Alphanumeric, dash and underscore but no spaces regular expression check JavaScript

However, the code below allows spaces.

No, it doesn't. However, it will only match on input with a length of 1. For inputs with a length greater than or equal to 1, you need a + following the character class:

var regexp = /^[a-zA-Z0-9-_]+$/;
var check = "checkme";
if (check.search(regexp) === -1)
    { alert('invalid'); }
else
    { alert('valid'); }

Note that neither the - (in this instance) nor the _ need escaping.

You shouldn't use String.match but RegExp.prototype.test (i.e. /abc/.test("abcd")) instead of String.search() if you're only interested in a boolean value. You also need to repeat your character class as explained in the answer by Andy E:

var regexp = /^[a-zA-Z0-9-_]+$/;

This is the most concise syntax I could find for a regex expression to be used for this check:

const regex = /^[\w-]+$/;