Hi, Jamie —
The most extensive relic collection in the United States is at:
It is probably the largest collection in the world that is open to public visitors. There is also a notable collection at:
In Chicago, the famous St. John Cantius Church has about 1,800 relics though only a small portion of them are on public display at any time.
For particular relics of interest, you may be able to find them by visiting a shrine or cathedral named in that saint's honor or founded by that saint. For example, the Cathedral of the Madeleine in Salt Lake City has a relic of St. Mary Magdalene. The body and heart of St. Andre Bessette (in separate locations) are in St. Joseph Oratory in Montreal.
I haven't investigated this yet, but you might find it worthwhile to look into dioceses that maintain their own historical museums or displays of archive material. (Every diocese has an archive department, but not all, have public displays.) If they have a museum open to the public, you might call to ask whether they have a relic collection displayed.
For example, the Los Angeles Archdiocesan Archival Center has a public display of relics, mentioned here.
Here is an organization devoted to spreading knowledge of and devotion to holy relics:
And here's an article about some of the collections I mentioned above:
I hope this is helpful.
Richard Chonak, for AskACatholic.com