I am often asked by new business owners whether they should pay to register their business name through ASIC or not. This costs $35 for a one-year registration and $82 for three years.
Previously, I have told my clients that it's not strictly necessary, but a good idea as it dissuades other businesses from choosing the same name (although doesn't stop them for choosing the same name - that would take a Trademark registration). Also, if you envision that you might float the company on the Stock Exchange at a later date, and sell public shares in the company, having the name registered with ASIC means that you have secured that name for the business.
However, changes are afoot. You are no longer able to simply add your business name to your ABN (i.e. as a Trading As name), you will need to register your business name with ASIC if you want it to show up on an ABN LookUp search.
For Sole Traders, this now means that only your own name (e.g. Mr. Brown) will show up in an ABN LookUp search, not your business name. As of November 2018, any previously connected Trading names will no longer be shown alongside your ABN. However, if you have registered your business name with ASIC, the records will be connected, and your business name will show up next to your ABN.
Here is the official page on this update.