Can a stoichiometric mixture of oxygen and methane exist as a liquid at standard pressure and some (low) temperature? This answer to the question Pre-mixing cryogenic fuels and using only one fuel tank written by a non-chemist (me) begins with:

At STP:

LOX's boiling point is 90.19 K
Methane's freezing point is  90.7 K

This does not a priori prove that a solution of the two can not exist. However it does mean that they can not be handled as liquids at the same temperature, making mixing the two more difficult.
We know that liquid air exists which shows that LOX and LN2 can mix together. But methane is an organic molecules and we know that heavier $\text{C}_n \text{H}_{2n+2}$ hydrocarbons include oils and waxes don't like to dissolve in non-organic solvents.

A stoichiometric mixture of oxygen and methane would be 2:1 molar:
$$\ce{ 2O2 + CH4 -&gt; CO2 + 2H2O }$$
Though the two can not be conveniently maintained as liquids at the same temperature, can a stoichiometric mixture of the two exist as a liquid at some (low) temperature and standard pressure?
There's a NASA report that looks into this: "ON THE SOLUBILITIES AND RATES OF SOLUTION OF GASES IN LIQUID METHANE", Hibbard and Evans, 1968 and concludes that such mixtures are possible.

Starting on page 8:


  Figure 5(a) presents the curves for oxygen, argon, carbon monoxide,
  and nitrogen. Also shown are the two experimental values for nitrogen.
  Agreement is excellent at 99.83K and good at 110.9K. The curves for
  these gases show that solubility should decrease with increasing
  temperature and the nitrogen data confirm this. This figure shows the
  mole fraction solubility of oxygen to be 1.0 at 90K. This means that
  oxygen, which has a normal boiling temperature of 90.1K would
  continuously condense in, and be miscible in all proportions, with
  liquid methane at 90K. This is confirmed by reference 11 where, in a
  study of the solubility of methane in liquid oxygen, it was concluded
  that these formed a near-ideal solution at -297 F (90K)


(emphasis added) Reference 11 mentioned in there is "Hydrocarbon-Oxygen Systems Solubility", McKinley and Wang, 1960 (unfortunately paywalled) which also has interesting discussion of the stability (i.e. presence or absence of a tendency to explode) of various mixtures.

Figure 5 is reproduced below. Note how the solubility of oxygen rises rapidly as temperature drops.