On (minuscule) dark matter production in supernovae It is believed that dark matter is made of particles, which interact with matter only weakly and gravitationally. One common candidate for dark matter are so called WIMPs. WIMPs, specifically, are heavy and may be their own antiparticles.

And as any other particles dark matter particles can be produced at sufficiently high energies. The mass of dark matter particles is unknown, but is estimated to be of order $1$-$100 \textrm{GeV}$, which corresponds to temperatures of $T_{DM}\approx 10^{13}$-$10^{15}\textrm{K}$, at which these particles may be expected to be produced.

Such enormous temperatures are barely attainable in any reasonable astrophysical processes, but say in core-collapse supernovae newly formed core has temperatures of $T_{SN,after}\approx 10^{11}\textrm{K}$, and probably more during the collapse phase. Then a crude estimate would suggest that the amount of dark matter produced is $M_{DM}\approx e^{-T_{DM}/T_{SN,max}}M_\odot$. Or, in number form $\log_{10}(M_{DM}/\textrm{kg})=30.3-0.43(T_{DM}/T_{SN})$. This means that at $T_{SN}=1.4\cdot 10^{-2}T_{DM}$ the amount of dark matter produced during a supernova will be around one kilogram. Such temperatures are fairly reachable for $1 \textrm{GeV}$ DM particles. So one can optimistically expect few kilograms of dark matter produced per supernova.

Now the question. What is a typical dark matter production in core-collapse supernovae? A good answer, I imagine, would be a more robust expansion on the existing estimate. Any constructive comments are welcome.
The most favoured WIMPS at the moment are probably neutralinos, see
http://en.wikipedia.org/wiki/Neutralino

These particles are purely hypothetical at the moment.
The mass estimates in the above Wikipedia article for the lightest neutralino range between 
10 and 10,000 GeV, meaning that the production rates in SNs will be much lower than with an assumed 1 GeV.
Higher production rates should have already been detected at LHC.

Hence from the non-detection (in the form of energy loss) of WIMPS at LHC an estimate of an upper bound of the production rates in SNs should be possible.