High angular resolution observations with the very long baseline interferometry (VLBI) provide crucial tests for the association of neutrinos with blazars. The VLBI follow-up observations have led to the unveiling of VLBI-scales properties potentially linked to the neutrino production in the jet of the blazar TXS 0506+056, the first source associated with a neutrino event. In order to collect new pieces of evidence on the connection between radio properties in blazars and neutrino production, we performed a VLBI investigation of gamma-ray blazars observed in spatial coincidence with four neutrino events, detected by the IceCube observatory between 2019 and 2020. We compared our new observations with archival data for selected candidate counterparts with the aim of searching for hints of a possible connection between the neutrino events and those
sources. This contribution presents some of the results of the first exploratory VLBI follow-up campaign, with a focus on the results of the most promising targets. A brief summary of the new follow-up observations for IceCube events carried out during 2022 is also given. Among these, there are two events, IC 211208A and IC 220205A, the candidate counterparts of which are flaring at the neutrino detection. They are the blazar PKS 0735+038 and the extremely peculiar source PKS B1413+135, respectively