We can’t take a survey of each and every Catholic diocese, parish, church, and chapel in the world, but sources like UCatholic state that all of them contain relics. While we don’t often see new churches being built, it stands to reason that the Vatican has some relics on standby to distribute as necessary.
That being said, Catholic Answers lives up to its name by answering a crucial question for the skeptical: the church has never officially endorsed a relic as genuine (despite apparent certificates of authenticity, as The Leaven describes). Theological particulars about the Intercession of the Saints notwithstanding, it’s at least true that the church recognizes the meaning that relics have for people, like holding onto a keepsake. This is probably a good perspective to adopt, because even in the modern day, supposed relics keep cropping up. Live Science, for instance, describes archaeologists in Turkey in 2013 claiming to have found a piece of Jesus’ cross in a stone chest in the basement of the 7th-century Balatlar Church in Sinop along the coast of the Black Sea.
UCatholic tells us that the practice of literally holding on to the pieces of the dead started with the Second Council of Nicaea in 787 C.E. The goal of the meeting was to unite the church and unify its still-developing doctrine, as Papal Encyclicals Online states. The council decided that — perhaps for the sake of continuity with Christian forbears — each church ought to have an “installation of relics.”