Famous phrase very suitable for this study, pronounced by one of the foremost experts on the Dead Sea Scrolls, the historian and philologist Geza Vermes (University of Oxford):

“The Dead Sea Scrolls allow us to hear for the first time, the authentic voice of judaism in the time in which it was born of christianity.”
— Geza Vermes (1924-2013), historian Hungarian-british, a pioneer in the study of Qumran.

1) Museums where to see them physically (originals and facsimiles)

Israel Museum – “Shrine of the Book” (Jerusalem)
It is the headquarters of the global reference. Houses and exhibits in rotation emblematic pieces (e.g., sections of the Great scroll of Isaiah) and facsimiles of other key texts; the curator explains the discovery and conservation. For reasons of preservation, the original is displayed for short periods of time and alternate with copies of high fidelity.

The Jordan Museum (Amman)
Exhibits the only Roll of Copper (inscription on copper foil of the s. I d. C.) in display case with climate control, in addition to materials associated with the findings under the management of jordan at the time.

Rockefeller Archaeological Museum (East Jerusalem)
Historically rather treasured fragments and was the center of editorial work for the rolls; today, its main role is proprietary and file, with the collection managed by the Israel Antiquities Authority (IAA), and transfers of parts to the Israel Museum. Useful for historical context of research and conservation.

Note of authenticity and ethics
In the last decade, several “fragments” acquired by private institutions were modern (not ancient). The most widespread was the Museum of the Bible (Washington, D. C.), which withdrew all his displays of fragments “of Qumran” as false, according to expert scientists. When planning a visit, check to see if the museum's exhibits original provenance and permissions IAA/Jordanor facsimiles.

2) Where to watch them online (open access, images and reading)

Leon Levy Dead Sea Scrolls Digital Library – IAA (free)
The official platform of the Israel Antiquities authority brings together thousands of fragments in image multispectral ultra-high resolution, tabs, techniques, and resource conservation. It is the standard for public viewing and academic open.

Digital Dead Sea Scrolls – Israel Museum (free)
Site of the Israel Museum project (with Google) with direct navigation of the five rolls “star”: Isaiah, Rule of the Community, Pesher Habakkuk, Temple scroll and Roll of the War; allows you to zoom end, and (in Isaiah) search column/chapter/verse.

Google Arts & Culture – Shrine of the Book (free)
Interactive stories and visualizations of the Great scroll of Isaiah, and of the building of the Shrine of the Book, useful for the general public and educational material.

Brill – Dead Sea Scrolls Electronic Library (institutional subscription)
Corpus-mail full (biblical and non-biblical) with transcripts in Hebrew and English translationsin addition of high-resolution images; it is the career path which is more robust to work philological if you have access to a university or college library.

3) Other ways to see them

(a) The archaeological site of Qumran (National Park)
There are No rolls on the side, but yes the caves, the ruins and the visitor center with introductory movie. It is the best complement to understand the landscape, the community and the context of the discovery.

(b) temporary Exhibits, and tours
Many museums have mounted temporary exhibitions in association with the IAA. If you are interested in a specific city, it is worth checking if the current sample presents original (with loan IAA or Jordan) or facsimiles.

c) critical Editions and translations
For systematic reading and study: the series DJD (Discoveries in the Judaean Desert) on paper, and electronic editions of Brill (see above) with the transcripts consolidated. For disclosure specialized, the Israel Museum and the IAA provide introductions to date.

d) Conservation and scientific photography
The IAA documents the program preventive conservation and digitization multispectral, key to understanding why many parts look better on the screen than to the naked eye in the room (rotations, sensitivity to light, etc).

4) Itinerary practical

• Original vs. facsimile: by rules of conservation, the original rotate and they are usually very little time; in thy visit to the Shrine of the Book you'll see often facsimiles of high-fidelity along with explanations curatorial.
• Authenticity: avoid plan visits based on “fragments” advertised by private institutions without official loan; the case of the Museum of the Bible made it clear the problem of the parts post-2002 of origin opaque.
• In-depth study: combine the experience of the room with the platforms IAA/Israel Museum to read, extend, and compare columns and palaeography with zoom (especially Isaiah 1QIsaa).