Depends on the design of a/c system.
One common design is, there are waste tanks for all lavatories.
Some use individual tanks immediately located under your 'seat'.
These will be serviced at each turnround by means of a fill & drain
system. A flush motor & adjacent timer forms part of the system.
They will have their own fill & drain service point. Older DC-10/BAe146.
The 'newer' type of system is the vacuum system. This vacuum
system is derived from the aircraft pressurisation or pneumatic system
but also uses the same bleach based medium to rinse. Gate valves & timer's again employed. The waste tank is located elsewhere ( underfloor/adjacent freight bays etc). Filled and drained in a similar way to above (Far less service points) Embraer190/AirbusA330.
Typically, the waste that is actually designed to be dumped
overboard in flight is lavatory & galley sinks through heated drain masts.
This explain's the coffee slicks that can be found immediately behind drain masts & the inevitable blockages that occur because Inflite Crumpet or
Pax pour glue down it (or equivalent of)..
I'm not rated on the A380 (Doesn't look like I will be for a while) But more than likely, the shower system certain customers may have installed will
probably also exit through these drain masts (Reduce weight where poss).
The 'blue ice' incidents are more about fill & drain leakage issues
and should be rectified promptly. Better individual system description can be found in an applicable Aircraft Maintenance Manual, Chapter 38 (Water/Waste).
BAe146???