Saburhane is the strongest reason to stop in Muğla. The quarter rises north-east of the centre in a tangle of lanes, whitewashed walls, tiled roofs, and 18th- and 19th-century houses that mix Ottoman and Greek domestic styles.
There is no single sight to tick off. Give it time, wander uphill, and let the old town do what resort towns on the coast rarely can: feel lived-in and unhurried.