Let's assume that I am dropped into a room with a Raspberry Pi running either Debian or Raspbian. How do I find out if it has hard float support or if it is just using soft float?
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Check for the existence of the directory:
/lib/arm-linux-gnueabihf
the soft-float version do not have this directory, they have:
/lib/arm-linux-gnueabi
instead, or you can list the packages installed using:
dpkg -l
and see the platform in the third column (all/armhf/armel)

ikku
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The accepted answer is correct, but a really simple method is:
dpkg --print-architecture
If your system is using a hard float architecture, you will then see:
armhf
ARM Hard Float. :-)

Chris
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Issue the command:
readelf -A $(which dpkg)
On a hard-float (hf) system, you will see this as part of the output:
Tag_ABI_VFP_args: VFP registers

Will
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1This also works on custom, non-Debian, non-Raspberry systems. I like that. – Johannes Overmann Dec 12 '18 at 08:59
/lib/arm-linux-gnueabihf
, your system is hard-float. – goldilocks Mar 25 '15 at 16:16