Hello, the topic is extremely wide. To start I could suggest a book, dated 2002 but still very valid,
Geotechnical Aspects of Landfill Design and Construction by Qian, Koerner and Gray, Pearson publisher.
In the last two decades many papers have been published on this subject, I suggest to look for main congress proceedings or journal selection.
Many factors are affecting characterization and stability: type of waste, age, moisture, compaction, dump construction, leachate and biogas management, presenze of geosynthetics, loads, geometry etc..
The physical stability of rubbish dump slope can be verified by performing slope stability analysis ..based on the analysis if factor of safety arrived by more than 3 or 5 then, the slope is stable otherwise unstable..the study to be conducted both field investigation and laboratory investigations for arriving the properties of rubbish dump to perform slope stability analysis..Bishop method is used to verify the stability of slope and many more methods are available ..i suggest you to read soil mechanics text book for various slope stability analysis method..ok all the best..
Depending upon the site and orientation of rubbish dump, You may assess its physical morphology and depending upon physical/site assessment and heterogeneous material, you may choose preferred locations/section lines, along which you may carry out slope stability assessment with a view to arrive at the status of slope in terms of FoS or Factor of Safety. When < 2, it is normally unstable/critically stable. You need to have the lab-estimated values of select geotechnical parameters used in a particular analysis method. Emperically chosen values from available datasets/ tables etc. may also be used for cross-check, but ideally lab-estimated values would be of more help.