There was a great deal of debate about the position of engines on jetliners in the 1950s.
The Avro Canada C102 Jetliner, the Comet and the TU104 had engines close to the wing root. The Comet's were more "buried" than the other two designs but engines close to the wing root - a design used by many large and small military aircraft - have some benefits over engines slung under the wings in terms of airflow and lack of FO damage. On the other hand they can be difficult to service, more susceptible to causing major, perhaps catastrophic, damage in the case of fire or turbine blade fragmentation and the noise and vibration in the cabin tends to be the greatest of all three engine positions.
Underslung engines (the design came out with the Me262 and was copied by the one off Tay engined Viscount) was adapted by Boeing to place the engines further below the wings by means of pylons - though they reverted to the Me262 position for the 737-100/200 srs.
The engines in this position are easier to service, are designed to break off if suffering catastrophic damage, are less likely to damage the empennage if they fail but have the disadvantage of spoiling the aerodynamic cleanliness of the wing.
Rear engines have the great advantage of leaving an aerodynamically clean wing which is important for short field performance - thus the application in the 1960s when jet engines were not as powerful for their size as they are today - and reputedly clean wing aircraft are more economic on fuel. BOAC wanted "hot and high" performance for the VC10. The Caravelle, Trident, 1-11, DC9, B727, F28 all needed to get into places where field length was restricted. All airlines putting rear engined jets into service made much of the lack of noise and vibration in the cabin.
The down sides of rear engines are many. More difficult to service (especially a buried centre engine). Extremely difficult, and therefore expensive in cost and weight, to prevent catastrophic damage to the airframe and systems passing the engines and the most important, the T tail deep stall, which at the cost of lives and airframes in the Trident and 1-11 testing programmes, only came to light after the design was built leading to the innovation of stick pushers.
The advent of more powerful small engines which could fit under the wing of medium sized jetliners saw the end of the rear engine except for the 1990s crop of regional jet designs where the same problem of engine size and power, linked to a small airframe, could only sensibly be overcome with rear engines. The same goes for a host of biz jets.
BTW, the three and a half engined Trident came about because the Trident 1 and 2 had poor climb performance (they were known as Grippers due to their unwillingness to leave the ground) and the larger size of the Trident 3 needed more power which the three Speys could not give. In practice, BEA/BA found little use for the extra engine on most of the Trident 3 routes and many were blanked off and I seem to remember most were removed.