We present four different constructions of pattern-avoiding inversion sequences, and use them to obtain expressions for the enumeration sequences of 22 distinct open cases. This completes the enumeration of inversion sequences avoiding one or two patterns of length 3. Some of our constructions are based on generating trees. Others involve pattern-avoiding words, which we also count using generating trees. Furthermore, we introduce a generalization of inversion sequences, which we call shifted inversion sequences. It is sometimes easier to construct pattern-avoiding inversion sequences by seeing them as a particular case of shifted inversion sequences.