The official Shroud site lists the expositions of the modern era, 1931, 1933, 1969, 1973, 1978, 1998, 2000, 2010, 2013 on television and 2015, and stated that none was planned for the 2025 Jubilee, offering instead a digital experience. Nothing has been published since for 2026 or 2027, and an exposition is normally announced a year or more ahead because the cathedral has to be refitted and the city prepares for millions of visitors. Watch sindone.org and the archdiocese for an announcement; speculation about 2033, the two-thousandth anniversary of the Passion, has not been confirmed by Turin.
What you can see now. In the cathedral, pray before the chapel where the case is kept and see the full-size photographic replica; the tomb of St. Pier Giorgio Frassati is in the same church. The Museum of the Shroud, in the crypt of the church of the Santissimo Sudario at Via San Domenico 28, is open every day 3:00 to 6:00 p.m. and holds the 1898 and 1931 photographs, the reliquary history and the scientific studies. The feast of the Holy Shroud is May 4, when the archbishop celebrates Mass in the cathedral without exposing the cloth.
The standing answer about display in general is on is-the-shroud-of-turin-on-public-display.