A large amount of dirt (rock and soil) would be dug up in forming trenched for HSR into and out of Melbourne, Sydney and Brisbane. It would need a means of disposal.
Soil could be dumped in out-of-the-way places. It would be used to make embankments where necessary. The amount of soil produced is likely to exceed these solutions.
Singapore is digging Metro tunnels and using the soil to enlarge the island. It pays to import soil from Indonesia. Singapore may very much be interested in importing large quantities of Australian excess soil. It is an expert in the use of spoil.
Australian HSR trench spoil is generated close to shipping ports and suburban railways in the major cities. It would be easy to export at low cost. It would need to be handled as little and as easily and quickly as possible. It may be better and cheaper not to handle it lose. Mostly existing infrastructure could be used. It could be dug and loaded direct into specially designed shipping containers, loaded onto freight trains next to the dig in non-operating passenger train hours on suburban tracks, moved to wharves and loaded onto container ships queued waiting to take them to Singapore for dumping. The containers and ships would return for more loads. This would avoid many road trucks moving the spoil through suburban streets day and night. Disposal of millions of tonnes would be readily achieved.
Alternatively, spoil could be used to raise the level of nearby land threatened by rising sea levels in low lying Australian coastal areas. It may be a slower redistribution method that would have to keep up with production rates.