Combustion in liquid state
Your question is addressed by Explosive Systems Containing Liquid Oxygen: Liquid Oxygen-Liquid Methane Mixtures. J. Chem. Eng. Data, 1959, vol. 4, pp. 127–131.

If liquid methane and liquid oxgygen at 90K are mixed, they are miscible in all proportions.  Mixtures over a wide range of proportions detonated with use of a blasting cap. 100 grams of mixture punched a 10cm hole in a 3/16 inch steel plate. 

For comparision, the TNT control only dented the metal plate!

Further tests on the stoichimetric liquid methane/oxgygen mixture found that it was detonated by a .22 caliber copper bullet at 40 foot range, flame and spark.

So nothing happens if the liquids are simply mixed, but a powerful explosion occurs if a stimulus is applied.  

Liquid oxygen and liquid methane have also been used in rocket engines.  See Liquid Methane/Liquid Oxygen Injectors for Potential Future Mars Ascent Engines for example.