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We had a similar issue a few months back but it occurred when we turned our hot water on. Turned out there was an issue with a valve in our hot water service, which required a visit from a plumber... Hope you can find a solution to your water woes!
im no pro when it comes to plumbing - but had a similar issue. "water hammer". could be because there is too much water pressure and/or your pipes are not fixed properly. if you can get access to the pipes you can check it and use saddles to fix the pipe more securely.
if your pipes are secured you may need to consider getting a plumber in to reduce the pressure for you.
alernatively if it is not a mixer tap - you can get tap valves that help with this problem at most hardware stores.
Is it only happening with one set of taps?. ie. Is it just the taps that go into your washing machine or dishwasher for example?. That would be water hammer and easy fixed with some anti water hammer valves which just screw in between you tap head and connecting pipe.
If it is happening to all taps and if you have the traditional spindle tap handles, water hammer won't be the issue as you aren't shutting off the water flow quickly.
May pay to get under the house or in the roof and get someone to replicate the issue. Could very well be a loose pipe.
All else fails, call in a plumber.
So helpful, thank you for this post! I have this problem since I got my new LG front load washer, it is only for the start of a wash, then it is fine. Reading your post I understand what is happening & will check out these anti hammer valves at Bunnings.