My jQuery Form Plugin
From a once over;


I would not split statements like 

this.$status.addClass('error')
    .text(message);


over 2 lines, I would reserve that treatment for the longer statements.
You seem to cheat with the simpleness of your call, Form knows already the error messages with this.errors and the validation types with this.required. These should have come in as options.
I tend to comment all my regular expressions with my intent and an example, that might be overkill for your regular expressions
In Form.prototype.process it would have been simpler to have var self = this and use self instead of using $.proxy(. While using closures is not a panacea, it should also not be avoided at all costs.
You seem to like shortcuts, consider

return filled = false;


instead of

filled = false;
return false;

filled &amp;&amp; this.submit($(this.$form).serialize()); &lt;-- Too hacky
Why is this code removing the shake css class, it never assigns it..
On the whole it seems Form.prototype.update could be written in a smarter way, but I would have to see an actual snippet working to get started on that