Comparing Multivariable-Regression Models Derived from Different Sample Sizes

I have a small dataset (n=39) with one dependent variable (y) and several independent variables (x1, x2, …, xn). For most of my independent variables I have 39 measurements. However, I am missing some measurements for some independent variables. For example, independent variable x1 is composed of 39 measurements but independent variable x2 and x3 are composed of 30 and 32 measurements, respectively.

I have built several simple- and multivariable-linear regression models to predict the dependent variable and I am having difficulty selecting the best model. Since some independent variables did not have 39 measurements, models were built different sample sizes. Most models had a sample size of 30. I had originally planned to use the corrected Akaike information criterion (AICc) for model selection (Wikipedia) but I recently learned that the AICc is only supposed to be used to compare models with the same sample size (Stack Exchange Link).

What model selecting criteria can I use to in this situation? I am using python for my analysis.

Multiple imputation of the missing data is one accepted way to approach this problem. There are nearly 400 questions on this site tagged as multiple-imputation. That's valid provided that the data are missing at random in a specific way, often a reasonable assumption.