Ferries
There are two scheduled boat services that connect Rum with the mainland:
The Caledonian MacBrayne (http://www.calmac.co.uk/) ferry MV Lochnevis sails to all four of the Small Isles from Mallaig throughout the year. It calls at Rum once on Monday, twice on Tuesday, twice on Thursday and once on Saturday during the winter; twice on Monday, twice on Wednesday, twice on Friday and twice on Saturday in the summer.
During the summer, Arisaig Marine (http://www.arisaig.co.uk/) provide wildlife sightseeing cruises from Arisaig, about 10 miles south of Mallaig. Different islands are called at on different days, but the timetable generally allows access to Muck on days when the CalMac ferry does not.
Train Connections
If not travelling to Mallaig or Arisaig by car, you can reach the ferries by scheduled ScotRail train service although note that the pier at Arisaig is not as close to the railway as the pier at Mallaig.
The famous ScotRail (http://www.scotrail.co.uk/) 'Deerstalker' Caledonian Sleeper provides first class single cabin and standard class double cabin sleeper and reclining seat travel between Fort William and London Euston every night except Saturday. Local trains connect to Mallaig.
If travelling by day train, travelling to Rum from anywhere further south than Fort William is only possible without an overnight stop in Mallaig on summer Saturdays, when the early morning train from Glasgow Queen Street station connects with the second CalMac sailing to Rum.
Travelling from Rum to points beyond Fort William by day train is likewise only possible on summer Saturdays, when the first CalMac sailing from Rum connects with the train to Fort William and Glasgow Queen Street.