After going over one of my older questions about forward swept wings, I realize I have never seen a forwards-swept horizontal stabilizer. Do they exist? What would be their advantages, if any?
Forward sweep is chosen when the designers value resistance against tip stall and a rear location of the wing spar higher than low mass. None of those advantages of a forward swept wing has any weight in the design of a horizontal tail, so a forward swept tail has never been selected.
The HFB Hansa , a business jet made in Germany in the 60s used forward swept wings, as did several research jets in the 60s -70s but I don't think any had a forward swept stabilizer. Forward swept lifting surfaces have the advantages of rearward swept surfaces, delaying shock wave formation to higher Mach number by reducing the effective thickness of the airfoil, without some of the disadvantages, tip first stall, decreased roll response, decreasing effectiveness at high angles of attack. But the forward swept lifting surfaces have disadvantages as well like, the requirements for much stiffer structure to avoid flutter and more tendency for Dutch roll.
With the advent of fly by wire, and the relaxed stability requirements that go along with it, the horizontal stabilizer on many new aircraft are operating at very low lift coefficients in cruise so I am not sure if there would be much advantage, except perhaps in a fighter optimized for gun fighting (close in high turn rate, heavily g loaded condition). The flutter issue is much easier to design for today, since we have extremely high modulus, low mass, graphite fibers we can use but nobody even seems interested in going with a forward swept wing, much less a stabilizer.
The aviation industry seems to have settled into a very conservative mind set after the heady days of the cold war, probably due to the immense cost of developing truly new technologies. I don't think we are likely to see a production forward swept, wing or tail, aircraft for quite some time.