It should be able to use regex when searching labels and statement values. e.g. inlabel:/(John|Jack)/ will match all items with labels containing word John or Jack. haswbstatement:P214=/@&~([0-9]+)/ will match all items with VIAF ID that is not number.
labels and descriptions are covered with the existing insource:// functionality, statements might be considered but wikidata already has indices that are almost too big to manage, and use more fields than elasticsearch supports. It's not likely we will be adding many more indexing methods to wikidata unless we end up buying a wikidata search cluster.
I suppose my best suggestion would be for wikidata to improve what is reported as the source text so that it covers the actual data in the item.