We review and investigate lepton flavor models, stemming from discrete non- Abelian flavor symmetries, described by one or two free model parameters. First, we confront eleven one- and seven two-parameter models with current results on leptonic mixing angles from global fits to neutrino oscillation data. We find that five of the one- and five of the two-parameter models survive the confrontation test at 3 sigma. Second, we investigate how these ten one- and two-parameter lepton flavor models may be discriminated at the proposed ESSnuSB experiment in Sweden. We show that the three one-parameter models that predict sin delta(CP) = 0 can be distinguished from those two that predict vertical bar sin delta(CP)vertical bar = 1 by at least 7 sigma. Finally, we find that three of the five one-parameter models can be excluded by at least 5 sigma and two of the one-parameter as well as at most two of the five two-parameter models can be excluded by at least 3 sigma with ESSnuSB if the true values of the leptonic mixing parameters remain close to the present best-fit values.