Aussie jobs data is here from earlier: Australian Employment Change (May report): +42K (vs. expected +10K)

Westpac's response is here: Australian jobs report - analyst responses

CBA's is here: Australian jobs report - analyst responses - part 2

More now, this via local Fairfax press collating:

David Gladwell, ANZ:

  • Encouragingly, average hours worked recorded a solid increase, and the underemployment rate also retreated from its recent peak. This will be a welcome result for households, with the absorption of spare labour capacity a prerequisite for any improvement in wages growth. We continue to believe that employment will rise from here, although the recent rapid growth is less likely to be sustained.

Rahul Bajoria, Barclays:

  • In terms of monetary policy, the RBA appears comfortable with a neutral stance for now, but ongoing concerns around the relative weakness in household consumption and wages will continue to keep the RBA watchful. We believe that, more than headline growth, if the trend of improvement in the labour market continues and wages start to rise at the margin, this may create conditions for the RBA to normalise policy, but not before those trends are firmly established. We think the RBA is likely to wait for CPI inflation to hold consistently above 2% over the next 6-12 months before signalling a shift in its stance.

Katie Hickie, Capital Economics:

  • The much stronger than anticipated rise in employment in May and the larger than expected fall in the unemployment rate will go some way to quashing growing talk of the chance of another interest rate cut by the RBA later this year. Nonetheless, there still appears to be plenty of spare capacity in the labour market, which will keep wage growth weak and mean that the RBA probably won't raise rates until 2019 either.

Craig James, CommSec:

  • One strong economic result is viewed with suspicion. Two strong results are viewed with cautious optimism. Three strong job results are viewed as confirmation of a very positive trend. And it makes sense. Job ads have been rising for some time and are at 6-year highs; business surveys show the best operating conditions in nine years; the global economy is lifting; and domestic profitability is at record highs.

Michael Turner, RBC:

  • Labour force reports tend to contain a high noise factor, but there is little point in denying the solid trend of the past three months. May's 42k increase in employment takes the 3m total up to 141k, which has dragged the unemployment rate down from 5.9% as of March to 5.5%. This purple patch of employment growth is given a little more credence by the alignment of some improving leading indicators, including job vacancies and surveyed expectations of headcount within firms.

(bolding above is mine)