I've been thinking about doing this recently.

I know you need a supporting lab as well for the application, so I was thinking along the lines of applying for funds jointly with my supervisors.

Does anyone know of people who have finished a PhD and then successfully applied for a grant to continue their own research?

I think this is a good idea and you should discuss it with professors and your supervisors in your department.

Hi TreeofLife,
I'm not in the nat sciences but I know a few of my mates who managed to continue at the uni post-PhD and applied for limited funding (i.e. funding covers research related issues but no stipend for personal expenditure etc…). I will try to get some details as I'm vague on the specifics.

Thanks Incognito. I also want to pay myself a proper postdoc salary as well though!

TreeofLife, I think funding options depend upon your field of study, publications of the investigators filing applications, and the quality of research proposal. I do not know much about it, but you may like to explore European Union FP7 program which I heard of from my professors in my university. It has quite good funding options. Apart from this, there may be good funding options for young researchers. Just check with UK research council and british council.

On a second thought, profile of the PI does play a significant role. And funding in some applications of your filed of study might be better than other. For example, if you can apply your research to some health related issue, you can get some funding from NHS as well. But do prepare a proposal and start applying. The proposal and documents you prepare for your first application might be used for applications for funding elsewhere.
Not a PhD yet and many of these things you might already be aware with, but have seen professors and some colleagues doing it and thought to share it with you :)

Hi TreeofLife,

yes, try to apply for post-doc funding, as academic jobs are scarce.

I hope I will finish with the write-up in the next few months, and I plan to use the time waiting for the viva/ doing corrections to write at least one post-doc proposal jointly with my supervisor.

Good luck

Is this what you did in the end ToL?