AAS Job Register: main source for astronomy job adverts; new postings on 1st of each month.
RAS Jobs emailing list: has some jobs not on AAS.
jobs.ac.uk - all UK academic jobs adverts.
RAS Fellowships. Note that only one RAS Fellowship can be held in a given University at one time - check the current list of Fellows to see where you can and can't apply.
Royal Society funding: Dorothy Hodgkin Fellowships, etc.
STFC Fellowships (>2yr post-doc experience required for Rutherford scheme; no post-doc fellowships since 2009/10).
Participants Portal for EU Horizon 2020 funding schemes. Some post-doc funding available (such as Marie Curie Fellowships).
EU "mobility" portal. Lots of jobs adverts, but not astronomy-specific.
Astronomy Jobs Rumour Mill. Gossip/info about who has been offered what. Also a reasonably comprehensive list of what jobs are available at present. (More reliable now that it used to be, as the wiki format tends to be self-correcting. Not always accurate though, and occasionally a bit crazy.)
Rhaana Starling has put together a very comprehensive list of post-doc fellowships here, as part of the Career Progression pages she maintains for the Equality & Diversity Working Group.
Other sources of advice
Sera Markoff gives a similar annual talk in Amsterdam - her slides are available on her website.
Matt Kenworthy (Leiden) has some words of advice for early-career researchers that should be required reading for anyone on the post-doc job market.
Brian Gaensler (Dunlap/Toronto) and Sarah Maddison (Monash) have compiled a detailed set of notes on job applications in academia (with an Australian focus, but lots of excellent general advice).
AstroBetter blog on writing fellowship applications.
Information on astronomy careers
RAS Report on astronomy careers in the UK (2005).
RAS Demographic Survey (2016).
AAS Careers in astronomy pages.
Science feature on careers in astronomy (2006).
UK-specific article from above.
"State of the profession" white papers, submitted to 2010 US Decadal Review. #51 (Seth et al.) and #69 (Zakamska et al.) are the most relevant here, containing recent info/stats on the US job market (as well as some interesting opinions).
Article on career stats in the US (presented at 2007 AAS meeting).
Compilation of AAS Astronomy Career Profiles.
Paul Crowther's "STFC Crisis" web-site. Scroll ~10% of the way down for data on UK post-doc numbers since 2003.
What Does a Successful Postdoctoral Fellowship Publication Record Look Like? - paper analysing publication records of current/recent NASA prize fellowship holders.