In Sybase, what can I do to figure out why my Transaction Log is filling up?
The transaction log is where Sybase stores all changes made to a database. It can be dumped separately from the database. Dumping the transaction log basically provides some kind of incremental backup functionality.

The first thing you need to figure out is whether you need the transaction log on this database i.e. is it OK for you if all data created since the last database dump are lost in case of a crash. If it is the case, a solution would be to enable truncation of the transaction log on checkpoint.
If you need the transaction log, you'll have to create a threshold action to dump it to a file. This will empty it and you can proceed.