There is no simple answer to the question of which aircraft types can use SEN and with what restrictions. It is necessary to look at the aircraft specific data in all cases. This is because within a generic aircraft type (e.g. A320) there can be lightweight and heavyweight versions, and subtypes operating at different engine thrust ratings. These differences can become significant on a performance limiting runway.
Many medium size jets will be takeoff weight limited at SEN, but the limitation is likely to manifest itself as either (a) full fuel, hence maximum range but restricted payload or (b) full payload but limited fuel which will constrain maximum range. I would expect scheduled operators to be most interested in (b), but one-off or occasional charters might accept (a).
So for example it may well be that a 737-800 could operate with a full payload on short sectors from SEN, but for longer sectors the likely payload restriction would make it a commercially unattractive option. This is why it is unlikely SEN would ever become a main base for this type.
The 757 has a very good airfield performance for an aircraft of its size. I would expect it to be able to operate to a good portfolio of European destinations from SEN but not a maximum payload transatlantic. That is a general comment based on my own experience, but again the real picture would only emerge from a proper performance calculation.