Is there any feature selection method specific for regression analysis?

Is there any feature selection method that works especially well for regressions?

I used backwards elimination and forward selection before a lot but I’ve recently read that even though it’s historical, it’s not correct… but maybe a variance threshold? Correlation threshold? Chi square? I’ve been searching but everywhere I look I find a new way of doing it… so I don’t get a definitive answer

Some context: I’m running a multinomial logit, with 3 classes on the y, and the regressors are a mix of demographics (age, sex, years at job, income, mortgage, debt to income ratio, etc), loan specific variables (loan amount, loan term, APR, rep APR) and some "time" variables (amount of days that client spent in certain status, how many times client entered the status). My objective is to predict.