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What do you mean exactly? How do you define 'unique'? And what opinions are you after? Route characteristics? Type of trains used, frequency of service? You need to be rather more specific.
Aren't they all unique? Ergo: there's only one branch line at that location?
Fawley Refinery branch. Unique because it operates only to serve the MoD Marchwood secure facility now (about a third of the way along the old branch); and, although it works "one train in section", it still has a fine complement of semaphores and gated level crossing
Well the Looe branch goes from a station/track which is aligned 90º from the main part of the station, the guard has to operate points on part of the line to allow the train to continue, interestingly once this is done the train essentially travels in the same direction as it just left but on a different lower section of track.
This thread has been locked until the OP returns to clarify what exactly he is asking for. Thread title has been corrected and thread moved to the correct forum.