Electrical and plumbing always; others depend on your state and the job's value. A plain-English map of Australia's trade licensing rules.

"Are they even allowed to do this job?" is a question more homeowners should ask — because for some trades, hiring an unlicensed operator isn't just risky, it can void your insurance and leave you with work that legally has to be redone.
The rules are state-based and genuinely messy, but the practical version fits in one article.
Electrical work and plumbing work require a licensed tradesperson in every Australian state and territory, full stop. That includes jobs that look small — replacing a power point, moving a tap, connecting a dishwasher's plumbing.
This is the one rule worth being strict about, because the failure modes are house fires and flooded kitchens, and insurers ask exactly one question when those happen.
Gas fitting requires licensing nationally. General building work typically requires a licensed builder once the job value crosses a state-specific threshold — in NSW, for example, residential building work over a few thousand dollars requires a licence, and several states licence waterproofing as its own class because getting it wrong is so expensive.
Air conditioning installation sits in an interesting overlap: the refrigerant handling side requires an ARC licence nationally, and the electrical connection requires an electrician.
Painting, tiling, landscaping, fencing, and general handyman work are unlicensed in most states for most job sizes (with exceptions — QLD licenses more trades than anywhere else, including painting above certain values). For these trades, your protection comes from reviews, references, insurance, and a clear quote rather than a register.
Every state runs a free public licence register — NSW Fair Trading, QBCC in Queensland, VBA in Victoria, and equivalents in the other states and territories. Search the licence number or name, confirm it's current, and confirm the licence class matches the work you're hiring for.
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