Even though we in non-rhotic-land don't pronounce the r at the end of a word like 'failure', we do sort of know that there is an r there, and if the following word starts with a vowel, we'll often pronounce it. So, "FailuRe is the best teacher" has the first r pronounced, and we usually don't say (though we might!) "Failya is the best teacha". This sort of r sound is called 'linking r'.
But perhaps we're not so good at knowing that 'failure' has an r at the end of it. 'Australia' doesn't, but that doesn't stop most of us from throwing one in there anyway if we have to say something like "Australia(r) is a big country". This is called an 'intrusive r', and it's basically the same phenomenon as linking r, just not quiiite as common because there are some people who pay careful attention to how words are written and have trained themselves to cut out of a lot of potential intrusive r's. (Rhotic speakers typically do not have intrusive r's in their speech – their brains know very well that 'failure' always ends in an r and 'Australia' never does, regardless of what sound follows the word.)
Intrusive r's in song give me a little hit of amusement when I hear them. I suspect they're rarer in singing than in speech, but I haven't done any sort of study on the subject (nor tried to look up if someone else has). Missy Higgins has a clear example in the chorus of Angela – "Aaangelaaa(r) Aaangelaaa" (first example at 0:37).
One immediately noticeable feature of Higgins is that she sings with an accent that's quite Australian. It's not as Australian as John Williamson's singing (or as English as Sophie Ellis-Bextor's), but it's clearly not the sort of generic pop singing accent that Australians would probably call American if asked to describe it, and IMO it gives her songs an extra earthy realness that they might not otherwise have.
One consequence (I think?) of identifying the generic singing accent as American is that non-rhotic speakers will often sing rhotically. Surprisingly enough, Americans often sing non-rhotically. For example, have a listen to Billy Joel in Piano Man: "It's nine o'clock on a Satuhday, the regulah crowd shuffles in." He pronounces a couple of rhotic r's but mostly omits them.
I don't want to suggest that this is a general rule of American singing voices! There are also singers like Billy Joel, whose Uptown Girl is 100% rhotic.
One feature of non-rhotic speakers imitating rhotic accents is that they often get the r's wrong, and become hyper-rhotic. The words 'pasta' and 'pastor' are non-rhotic homophones, as are 'sauce' and 'source', and it's very easy to imagine that Americans might therefore talk about pastar sources.
Before starting to write this post I vaguely recalled there being a famous example of the Beatles with an intrusive r or something, and it turned out to be a hyper-correction of this form. The song is a cover of Till There Was You, and Paul sings (0:40) "I never sore them winging". Somewhat remarkably, I'm either so bad at concentrating on song lyrics or so unattuned to hyper-rhoticity that, despite knowing the line I was listening for, I didn't hear it on the first listen and had to go back to verify.
This whole post was prompted by Rhythmbox shuffling onto Missy Higgins' They Weren't There, when I was paying enough attention to hear (1:28) "where any preconceived idears were blown away".
I was tempted to conclude by adapting the lyrics to the chorus of Scar, but I decided it'd be too forced.