myArray.sort(
  function(a, b) {
    if (a.toLowerCase() < b.toLowerCase()) return -1;
    if (a.toLowerCase() > b.toLowerCase()) return 1;
    return 0;
  }
);

编辑: 请注意，我最初写这篇文章是为了说明技术，而不是考虑性能。也请参考回答@Ivan Krechetov更紧凑的解决方案。

如果你在努力理解以下内容，这可能会有所帮助:

var array = ["sort", "Me", "alphabetically", "But", "Ignore", "case"];
console.log('Unordered array ---', array, '------------');

array.sort(function(a,b) {
    a = a.toLowerCase();
    b = b.toLowerCase();
    console.log("Compare '" + a + "' and '" + b + "'");

    if( a == b) {
        console.log('Comparison result, 0 --- leave as is ');
        return 0;
    }
    if( a > b) {
        console.log('Comparison result, 1 --- move '+b+' to before '+a+' ');
        return 1;
    }
    console.log('Comparison result, -1 --- move '+a+' to before '+b+' ');
    return -1;


});

console.log('Ordered array ---', array, '------------');


// return logic

/***
If compareFunction(a, b) is less than 0, sort a to a lower index than b, i.e. a comes first.
If compareFunction(a, b) returns 0, leave a and b unchanged with respect to each other, but sorted with respect to all different elements. Note: the ECMAscript standard does not guarantee this behaviour, and thus not all browsers (e.g. Mozilla versions dating back to at least 2003) respect this.
If compareFunction(a, b) is greater than 0, sort b to a lower index than a.
***/

http://jsfiddle.net/ianjamieson/wmxn2ram/1/

如果不管输入数组中元素的顺序如何，你都想保证相同的顺序，这里是一个稳定排序:

myArray.sort(function(a, b) {
    /* Storing case insensitive comparison */
    var comparison = a.toLowerCase().localeCompare(b.toLowerCase());
    /* If strings are equal in case insensitive comparison */
    if (comparison === 0) {
        /* Return case sensitive comparison instead */
        return a.localeCompare(b);
    }
    /* Otherwise return result */
    return comparison;
});

我把上面的答案包装在一个填充中，这样我就可以在字符串数组上调用.sortIgnoreCase()

// Array.sortIgnoreCase() polyfill
if (!Array.prototype.sortIgnoreCase) {
    Array.prototype.sortIgnoreCase = function () {
        return this.sort(function (a, b) {
            return a.toLowerCase().localeCompare(b.toLowerCase());
        });
    };
}

在(几乎:)一行代码中

["Foo", "bar"].sort(function (a, b) {
    return a.toLowerCase().localeCompare(b.toLowerCase());
});

结果是

[ 'bar', 'Foo' ]

而

["Foo", "bar"].sort();

结果

[ 'Foo', 'bar' ]

myArray.sort(
  function(a, b) {
    if (a.toLowerCase() < b.toLowerCase()) return -1;
    if (a.toLowerCase() > b.toLowerCase()) return 1;
    return 0;
  }
);

编辑: 请注意，我最初写这篇文章是为了说明技术，而不是考虑性能。也请参考回答@Ivan Krechetov更紧凑的解决方案。