Crosswind landings
As the CFI of a busy PPL training school in the North of England, currently training 100 students, with only one runway, crosswind landings are a necessary part of student training otherwise one would never fly. Our flying order book states a crosswind limit for students of 12 knots for the PA28 and 10 knots for the 152. I am happy for students to fly solo upto the limits if they are capable, some are more than capable of exceeding those limits, however you would be on a hiding to nothing if you were to risk it and something went wrong. If they are not capable of managing the full allowable limit then the limit is lower, it all depends on the student.
Students inevitabley experience higher crosswinds when flying dual and will hopefully give them the necessary experience to deal with the higher crosswind limits they will be allowed to fly upto once in possession of their licence. As said on other posts it is wise to build up your crosswind limit bit by bit or fly dual until you feel confident.