I recommend multinomial logit or loglinear regression.

Multinomial logistic regression is based on the assumption that the DV is a nominal variable (discrete and unordered).  If your DV is ordered, you could use an ordered logit or probit model.  This produces fewer coefficients to interpret than a multinomial logit, but also assumes that the effect of a change in an IV is the same for all of the categories of the DV (known as the proportional odds or parallel lines assumption).

I hope this helps

Chris

In case you consider a nonparametric alternative:

• https://odajournal.com/2016/09/21/novometrics-vs-regression-analysis-modeling-patient-satisfaction-with-care-received-in-the-emergency-room/
• https://odajournal.com/2016/09/19/novometric-analysis-with-ordered-class-variables-the-optimal-alternative-to-linear-regression-analysis/
• https://odajournal.com/2017/04/18/what-is-optimal-data-analysis/