First, you have to consider the fact that not every organic molecule destabilizes under similar chemical conditions. To find a oraganic chemical trace of a fossil is rare, but provides informations on the morphology of the organism with graet fidelty.

After deposition/burial, the organic molecules (especially the least stable ones like protein, carbohydrates) gets degraded due to aerobic (and gradually anaerobic) bacterias. Proteolysis, or protein degradation is a set of processes that result in the hydrolysis of one or more of the peptide bonds in a protein, either through catalysis by proteolytic enzymes called proteases or nonenzymatically, for example at very low or high pH, T etc.

To preserve any animal Fossil, you need an environment with least "problems" like degradation, deformation etc. When the conditions are conducive, most of the soft parts gets leached into the soil/rock underneath and the "vacant" parts gets filled up with stable minerals (like pyrite, calcite, silica). Sometimes, the conditions get favourable for soft parts to get preserved also.

Now, Aromatic steranes are another group of biomarker compounds that are highly resistant to biodegradation and can be used for oil-to-oil correlation and oil source tracking. This can be seen for Cryogenian sponges.

Existem vários tipos de fossilização, os mais comuns são carbonificação, carbonatação, silicificação ...