Simplest way is to check voltage with and without engine running. In round terms a good battery will show in the order of 12 volts engine off and with the engine running this will increase up to just under 15volts.
Don't forget other factors such as corroded cables, etc.
Use a hydrometer. If any cell shows a different reading from the others, by two divisions on the scale, whether the battery is charged or not, it is duff.
Get a multimeter and read the output from the alternator.
I recently had a battery that I thought was iffy. So I removed it and took it to the local marine electrical chaps so they could measured the CCA (Cold Cranking Amps). I'm told that this is a good test.
This battery turned out to be okay, despite looking iffy on my hygrometer.
Yes, I think your experience is common - hydrometer readings are only completely useful if they can be compared with ones taken when the battery was first purchased. The SG will differ among batteries according to how they were provisioned with acid before sale (and measurement is further complicated by temperature).
Normally failure of a battery is quite obvious from its diminished or lost cranking ability or loss of capacity if in a service bank.
a) LED battery / alternator checker .... about 5 - 10 quid in Halfords
b) Hydrometer
a) will tell you if alternator is charging well ...... and also if battery is charged.
b) will tell you if the acid level is ok in the battery - which relates to charge level.
I have both and item a) was very useful in telling me alternator was goosed ... took alt. off - gave it to service guy who changed diodes etc. and it was back charging like a good'un. After repair re-checked with a) and it showed correct.
Once alternator is confirmed as pumping out ok - and you want to really check battery - local service should be able to put load tester across the battery. You could of course buy a tester ... there are now good DIY digital load-testers avialable that load-up batterys ... about 29 quid ... more if you feel inclined to spend more ...
A lot of "experts" will say forget the Halfords / Cheapo tester .... but it works - gives simple quick test that certainly worked for me.
Of course the hydometer only works with non sealed batteries!! Check alternator with multimeter, then if thats OK get the battery off and have it charged/tested by an expert, with a deep discharge tester, that will tell you the truth.
I have a domestic battery that I suspect is shagged. (Lights dim and radio goes off when other lights are turned on after an hour of running the engine).
The engine is only 250 hours old so it shouldn't be the alternator although once I have a multimeter I will test both.
Can anyone recommend a good supplier for a replacement?
batteries4leisure.co.uk Excellent service, recognised manufacturers and shipped within UK. I saved 40% over what one of the bid "marine" suppliers was quoting.