To just merge the arrays (without removing duplicates)

ES5 version use Array.concat:

var array1 = ["Vijendra", "Singh"];
var array2 = ["Singh", "Shakya"];

array1 = array1.concat(array2);

console.log(array1);

2023 update

The original answer was from years ago. ES6 is fully supported and IE is finally dead. Here's the simplest way to merge primitive and object arrays:

const merge = (a, b, predicate = (a, b) => a === b) => {
    const c = [...a]; // copy to avoid side effects
    // add all items from B to copy C if they're not already present
    b.forEach((bItem) => (c.some((cItem) => predicate(bItem, cItem)) ? null : c.push(bItem)))
    return c;
}

merge(['a', 'b', 'c'], ['c', 'x', 'd']);
// => ['a', 'b', 'c', 'x', 'd']

merge([{id: 1}, {id: 2}], [{id: 2}, {id: 3}], (a, b) => a.id === b.id);
// [{id: 1}, {id: 2}, {id: 3}]

Original answer

ES6 version use destructuring

const array1 = ["Vijendra","Singh"];
const array2 = ["Singh", "Shakya"];
const array3 = [...array1, ...array2];

Since there is no 'built in' way to remove duplicates (ECMA-262 actually has Array.forEach which would be great for this), we have to do it manually. Note that this pollutes the Array prototype, use with caution.

Array.prototype.unique = function() {
    var a = this.concat();
    for(var i=0; i<a.length; ++i) {
        for(var j=i+1; j<a.length; ++j) {
            if(a[i] === a[j])
                a.splice(j--, 1);
        }
    }

    return a;
};

Then, to use it:

var array1 = ["Vijendra","Singh"];
var array2 = ["Singh", "Shakya"];
// Merges both arrays and gets unique items
var array3 = array1.concat(array2).unique(); 

This will also preserve the order of the arrays (i.e, no sorting needed).

Since many people are annoyed about prototype augmentation of Array.prototype and for in loops, here is a less invasive way to use it:

function arrayUnique(array) {
    var a = array.concat();
    for(var i=0; i<a.length; ++i) {
        for(var j=i+1; j<a.length; ++j) {
            if(a[i] === a[j])
                a.splice(j--, 1);
        }
    }

    return a;
}

var array1 = ["Vijendra","Singh"];
var array2 = ["Singh", "Shakya"];
    // Merges both arrays and gets unique items
var array3 = arrayUnique(array1.concat(array2));

For those who are fortunate enough to work with browsers where ES5 is available, you can use Object.defineProperty like this:

Object.defineProperty(Array.prototype, 'unique', {
    enumerable: false,
    configurable: false,
    writable: false,
    value: function() {
        var a = this.concat();
        for(var i=0; i<a.length; ++i) {
            for(var j=i+1; j<a.length; ++j) {
                if(a[i] === a[j])
                    a.splice(j--, 1);
            }
        }

        return a;
    }
});

JavaScript engines typically put a cap on the number of arguments you can pass to a method before they throw an error, for example, the below throws an error in Chrome:

const arr = Array(150000).fill(0);
const arr2 = [];
arr2.push(...arr);

Whereas when using .concat(), you'll only be passing the one array to the method, so you don't get the error:

const arr = Array(150000).fill(0);
const arr2 = [];
const res = arr2.concat(arr);
// res is an array with 150000 `0`s

Additionally, with .push() + ..., you're effectively doing two iterations over your iterable/array, one for unpacking its contents as arguments to the .push() method, and then another for one done internally by the .push() method itself to iterate through each of the arguments and append it to the target array.

Another noticeable difference is in what both methods return, .concat() will return a new array and won't modify the target, which can be useful in methods such as .map() or .reduce() where you need to produce a new array without mutating the original. Whereas .push() will return the length of the updated array and will modify the target, so that is another difference to consider.

As pointed out by @T.J. Crowder, the iterator of arrays which is invoked when using the spread syntax ... does not preserve blanks in sparse arrays, instead it unpacks them as undefined values, meaning that if the array you're specifying is sparse when using .push() + ... you'll get undefined for the blanks, whereas the blanks will be preserved when using .concat() directly:

const arr = Array(3); // this is a sparse array, [empty × 3], not to be confused with [undefined, undefined, undefined]
const arr2 = [];
arr2.push(...arr); // Here `...` becomes: arr2.push(undefined, undefined, undefined);
console.log(arr2); // [undefined, undefined, undefined]

const arr3 = [];
console.log(arr3.concat(arr)); // Retains empties: [empty × 3]

See browser console for results