The paper is devoted to restricted Oppenheim expansion of real numbers ($ROE$),which includes as partial cases already known Engel, Silvester and Lüroth expansions. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $ROE$-expansion contain arbitrary digit $i$ only finitely many times. Main results of the paper states the singularity (w.r.t. Lebesgue measure) of the distribution of random variable with i.i.d increments of symbols of restricted Oppenheim expansion. General non-i.i.d. case are also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.