Definitely speak to ACAS. Regardless of the size of the organisation they should have policies. It is just bad manners and not appropriate for a partner to pull you up unnecessarily about your phone manner, and certainly your hair!
That said, you would expect a receptionist to be well groomed and good on the phone - do you recall a day when you had a difficult or rude customer? Or a particularly bad hair day?
Did they give you a formal warning? There are rules about warnings - you need to be told in advance and in writing that you are called to a disciplinary hearing, you can be accompanied by a colleague. They should then adjourn the meeting whilst they discuss the appropriate way forward.
You could make a formal complaint to one of the other partners, but if it's a small company you probably don't want to rock the boat. However, if it continues it could be construed to be harassment or bullying - but again if you make a formal complaint you have to accept the consequences and atmosphere that this could create. You'd need to be prepared to jump ship - and claim constructive dismissal if they make your life intolerable that you feel you can work there any more.
I friendly word to a senior person you get on well with us probably the best step forward.
Good luck, and let us know how you get on x
