This is a non-exhaustive list. The economics of open access publishing are evolving; note that article processing charges are not unique to open access publishing - reputable biomedical titles in particular have long used them for subscription journals. Long-established conventional journal publishers are expanding their reach into open access, with new titles, or moving old titles into open access, or hybrid titles with some articles open access and others behind a paywall.
Public Library of Science Founded in 2000 by Patrick O. Brown, Harold E. Varmus, & Michael Eisen. Megajournal PLOS ONE accepts original research in all scientific disciplines, including interdisciplinary research, negative results and replication studies – all vital parts of the scientific record.
Elsevier's Open Access journals (Authors pay article processing charges, access is free to readers). Heliyon is a completely open access megajournal from Elsevier that publishes robust research across all disciplines.
Springer's Open Access journals. (Authors pay article processing charges or their institutions pay membership fees, access is free to readers).
Nature Communications (megajournal) publishes high-quality papers from all areas of science that represent important advances within specific scientific disciplines, but that might not necessarily have the scientific reach of papers published in Nature and the Nature research journals.
Scientific Reports (megajournal) is an online, open access, unbranded journal from the publishers of Nature, which publishes primary research from all areas of the natural and clinical sciences.
Directory of Open Access Journals (DOAJ) Well monitored and curated list of reputable peer-reviewed open access journals.
CrossRef maintains a list of member journal titles for which it provides linking infrastructure services. The mission of this non-profit is "to enable easy identification and use of trustworthy electronic content". It is known to remove journals of dubious reputation from membership and thus from the list, so presence of a title on this list may be useful positive indicator.
Ulrich's Directory lists and describes traditionally published and open access journal titles. It can tell you whether a journal uses peer review, what discovery tools/databases index it, etc.
Investigating journals: the dark side of publishing. The explosion in open-access publishing has fuelled the rise of questionable operators. (27 March, 2013). Nature - News & Comment. (part of a Nature special issue on the future of publishing).
M. Pai and E, Franco (2016, October 4). What Are Predatory Open Access Journals And Why Should We Worry? Huffington Post Business Canada. (on predatory publisher OMICS buying Canadian medical journals)
Predatory journals recruit fake editor. Nature, March 22, 2017.
Beall's List of "potential, possible, or probable predatory scholarly open-access publishers". Beall is a librarian at the University of Colorado who closely monitored the seedy side of open access publishing.
From the original site:
"This is a list of questionable, scholarly open-access publishers. We recommend that scholars read the available reviews, assessments and descriptions provided here, and then decide for themselves whether they want to submit articles, serve as editors or on editorial boards. The criteria for determining predatory publishers are here.
We hope that tenure and promotion committees can also decide for themselves how importantly or not to rate articles published in these journals in the context of their own institutional standards and/or geocultural locus. We emphasize that journal publishers and journals change in their business and editorial practices over time. This list is kept up-to-date to the best extent possible but may not reflect sudden, unreported, or unknown enhancements."