The source of methane -- degradable organic carbon
A 2006 report by the IPCC, Guidelines for National Greenhouse Gas Inventories, Volume 5 -- Waste, includes six different categories of landfilled waste which contain degradable organic carbon (DOC). Under anaerobic conditions, the DOC in these materials ultimately decomposes into carbon dioxide (CO2) and methane (CH4). The eight categories are listed below, with the fraction (by weight) of DOC generally in each. Finally, I included the percent of landfill material that each category constitutes in North America (chosen since you reference California in the question -- data for other regions is available in the spreadsheet tool).
CATEGORY % DOC % LANDFILL COMPOSITION
----------- ----- ----------------------
Food waste 0.15 0.34
Garden 0.20 0.00
Paper 0.40 0.23
Wood and straw 0.43 0.06
Natural textiles 0.24 0.04
Disposable nappies 0.24 0.00
(Some garden waste is included with food waste. Nappies (diapers) have a value of 0 for all regions -- I'm not really sure why they aren't included, or what assumptions are made that allow them to be excluded here.)
Based on these assumptions, food waste and paper/cardboard are the largest sources of degradable organic carbon in landfills.
How much of it becomes methane?
The amount of emissions from the decomposition that becomes methane depends on the type of landfill and how it is managed:
Waste disposal practices vary in the control, placement of waste and management of the site. The CH4 correction factor (MCF) accounts for the fact that unmanaged SWDS [solid waste disposal sites] produce less CH4 from a given amount of waste than anaerobic managed SWDS.
The MCF by SWDS types are:
In a managed, anaerobic landfill, all of the DOC decomposes into methane. This is the "gold standard" in landfills, and in the U.S., the EPA requires most large landfills to collect this methane, where it is typically burned to produce electricity.
How much of the methane is released to the atmosphere?
The amount of methane captured at landfills varies widely, but in practice rates of up to 90% are feasible:
There have been some measurements of efficiencies at gas recovery projects, and reported efficiencies have been between 10 and 85 percent. Oonk and Boom (1995) measured efficiencies at closed, unlined SWDS to be in between 10 and 80 percent, the average over 11 SWDS being 37 percent. More recently Scharff et al. (2003) measured efficiencies at four SWDS to be 9 percent, 50 percent, 55 percent and 33 percent. Spokas et al. (2006) and Diot et al. (2001) recently measured efficiencies above 90 percent.
Bonus Q: How much food is wasted?
The assumption is that 34% of what is sent to landfills is food waste. How does this compare to the amount of food produced?
From a 2010 article in Scientific American:
According to the U.S. Department of Agriculture’s Food Loss Project, we throw away more than 25 percent—some 25.9 million tons—of all the food we produce for domestic sale and consumption. A 2004 University of Arizona study pegs the figure at closer to 50 percent [...]