To my knowledge there is no search option that appears on the OOTB To-Do page, nor does it have any filtering or categorization.
That is not to say you couldn't add this onto your page if you so desire, or raise an Idea through the Idea portal so it can be included as an OOTB option.
In a quick example, below I added typeahead search widget to the page. Can then go and setup a search context (which the widget needs defined) to return results based on the search context. Have a look at some of the existing OOTB search contexts to see how they are structured
