Recent success of coherent elastic [Nuclear Resonant Scattering of Synchrotron Radiation, Part A edited by E. Gerdau and H. de Woard (Baltzer Science, 2000), Hyperfine Interact. 123/124, Chap. 4] and incoherent inelastic (Hyperfine Interact. 123/124, Chap. 5) Mössbauer scattering of synchrotron radiation (SR) in investigations of very delicate properties of the condensed matter also makes it urgent to perform experiments on coherent inelastic Mössbauer scattering (CIMS) of synchrotron radiation (the common meaning of the term CIMS is coherent inelastic Mössbauer scattering accompanied by creation or annihilation of phonons in the crystal lattice, i.e., by very low energy losses of SR quanta). However up to now there were no publications on experimental observation of CIMS so there is a need in theoretical investigations to reveal the most favorable conditions for CIMS observation. The theory of CIMS is presented below and applied to specific processes of CIMS such as forward scattering, scattering at grazing incidence angles, and scattering via a cascade of Mössbauer transitions. It is shown that the phase matching (between the incident and scattered beam) is very important for the angular and frequency distribution in CIMS and processes where phase matching can be reached, which the best candidates for CIMS experimental investigations. The performed analysis shows that because of the phase matching demands the forward CIMS is suppressed significantly in comparison with the coherent elastic Mössbauer scattering [V. A. Belyakov, JETP Lett. 67, 8 (1998)] and more favorable for observation is CIMS at a nonzero scattering angle. Some examples of CIMS specific geometries are discussed. In particular, it is shown that for the grazing CIMS at isotope interface (a plane interface between regions with different abundance of the Mössbauer isotope) there is enhancement of CIMS at the critical angle of total reflection and suppression of CIMS at angles below the critical one [V. A. Belyakov, JETP Lett. 68, 287 (1998); V. A. Belyakov and S. V. Semenov, JETP 90, 290 (2000)]. Another possibility of CIMS in a more general meaning of the term is Mössbauer scattering of SR via a cascade of Mössbauer transitions (CIMC) which is connected with huge losses of energy by SR quanta in the process [V. A. Belyakov and Yu. M. Aivazian, Nucl. Instrum. Methods, Phys. Res. A 359, 190 (1995); 448, 222 (2000)]. Analysis of CIMC for the two transitions cascade in Fe57 and the corresponding calculations are presented. Optimal conditions of experimental observation for various cases of CIMS are discussed.