The ASME citation style mostly follows the Chicago Manual of Style. The info below comes from the ASME general overview of their citation style:
Book or eBook
Author, A.A., Author, B.B., and Author, C.C., Year, Book Title. Publisher Name, Publisher Location, DOI.
Example: Smith, J., 2014, A Dog’s Life in Berlin. Oxford University Press, New York. DOI: 10.1055/acprof.oso/97890.0394.000.
Note: A book will only have a DOI (Digital Object Identifier) if it's accessed online.
Chapter within a book or ebook
Author, A.A., Author, B.B., and Author, C.C., Year, "Chapter Title," Book Name. A.A. Editor, B.B. Editor. and C.C. Editor, eds., Publisher Name, Publisher Location, page numbers.
Example: Stevens, T. T., 1999, “Stochastic Fields and Their Digital Simulation,” Stochastic Methods. T. A. Sulle, and M. Siiu, eds., Martinius Publishers, Dordrecht, Germany, pp. 22-36.
Article from a Scholarly Journal
Author, A.A., Author, B.B., and Author, C.C., Year, "Article Title," Journal Name, Volume Number(Issue Number), Pages.
Example: Zhang, T. W., Khun, C., Liu, Q., and Miller, A. P., 2011, “Self-Healing Techniques,” Nature, 332(6662), pp. 888-892.
Author, A.A., Author, B.B., and Author, C.C., Year, "Conference Proceeding Title," Conference Proceedings Name, Conference Location, Conference Date, Conference Proceeding Number, Pages.
Example: Wions, T. T., and Mills, C. D., 2006, “Structural Dynamics in Parallel Manipulation,” Proceedings of the IDETC/CIE, New Orleans, LA, September 10-13, 2005, ASME Paper No. DETC2005-99532, pp. 777-798.
Author, A.A., Author, B.B., and Author, C.C., Year, "Title of Technical Report," Agency Name Report Number.
Example: Oligaria, T. T., Fredy, C. W., Popullo, A. Z., and Tucker, M. A., 2011, “Characterization of PKM Dynamics,” SAE Technical Paper No. 2011-02-8345, 07ATC-96.
Inventor, A.A., Inventor, B.B., and Inventor, C.C., Patent Name. Patent Number (including country), filing date, and issue date.
Example: Iizuka, M., and Hideki T. Cement admixture. US Patent 4,586,960, filed June 26, 1984, and issued May 6, 1986.
Name of standard granting organization. Title of standard. Standard number. Publisher location: Publisher name. Date approved.
Example: National Information Standards Organization. Bibliographic References. ANSI/NISO Z39.29-2005. Bethesda, MD: NISO, approved June 9, 2005; reaffirmed May 13, 2010.
"Website Title," Title/description of website as a whole, Owner/organization Name, Date of Publication or Revision, URL.
Example: “Apps for Office Sample Pack,” Office Dev Center, Microsoft Corporation, updated October 20, 2015, https://code.msdn.microsoft.com/office/Apps-for-Office-code-d04762b7.
Note: Provide as much information as possible. A date of publication or revision is preferred. If not available, use date of access.
Example references from: ASME Journals Digital Submission Tool Guidelines and Information - References and The Chicago Manual of Style.