The Psak that I received was that unmarried girls should absolutely light the Menorah – and not as “chinuch”, but as the basic “Mehadrin min haMehadrin” according to the Ashkenazi view. To recap the levels described for the Mitzva:
1. Basis – one candle each night for the whole household.
2. Mehadrin – one candle each night for each member of the household.
3. Mehadrin min haMehadrin (Ashkenazi) – each member of the household lights a number of candles equal to the night of of Chanuka.
Based on the above, the Mehadrin and Mehadrin min haMehadrin require EACH MEMBER of the household – whether male or female – to light. There is grounds for a married woman to not light if her husband does on account of “ishto kegufo”, but that only applies to a married woman – single women are no different from men with respect to this particular Mitzva. Accordingly, all of my daughters light, but my wife does not.
an Israeli Yid