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Here are the key information resources for locating spectra, as well as educational resources like textbooks.
Key Databases for Locating Spectra
SpectraBase (IR, M, NMR, R, UV)
Free databases of hundreds of thousands of spectra, including NMR, IR (including ATR-IR), Raman, UV-Vis, and Mass. Search by chemical name, CAS-RN, or InChiKey. You can upload and overlay your own spectrum for comparison. [Free, but registration is required. Some ads as well.]
SciFindern (IR, M, NMR, R, UV)
Largest online collection of spectra available for UCSD: 1.4 million experimental (mostly MS, IR, 13C-NMR, H-NMR) spectra for 983K compounds, and 109 million predicted H-NMR and 13C-NMR spectra.
Use the "Substance Identifier" option to search for your chemical, then from the substance (CAS Registry) record, look for the Experimental Spectra list. A "See Spectrum" link means the spectrum is available. Otherwise you will be directed to an article where you'll find the spectrum and/or numeric data. First-time users will need to register for a SciFinder account.
Reaxys (IR, M, NMR, R, UV)
Literature references for spectra for many of the 32 million substances indexed. From the substance record, select Spectra for NMR, IR, Mass, UV/VIS, Raman, Fluorescence and more.
There are no actual spectra, but you will get references to journal articles where you will find the spectrum and/or numeric information (peaks, shifts, etc).
PubChem (IR, M, NMR, R, UV)
Open database of millions of chemical substances, with spectral information for about 300K substances (found in the Chemical and Physical Properties section of the substance record). The data is sourced from several organizations, primarily SpectraBase, the NIST/EPA/NIH Mass Spectral Library, the Hazardous Substances Data Bank (HSDB), and the Human Metabolome Database (HMDB).
SDBS - Spectral Database for Organic Compounds (IR, M, NMR, R)
Spectra for 34,600 organic compounds, mostly FT-IR (54K), MS (25K), 13C-NMR (14K), and H-NMR (16K), and Raman (3.5K). Good for an initial quick search.
NIST Chemistry Webbook (IR, M, UV)
16K gas phase IR spectra, 15K mass spectra, and 1.6 UV/Vis spectra.
Properties of Organic Compounds (IR, M, NMR, R, UV)
Property data and numerical NMR, MS, UV and IR peaks for 29,000 organic compounds, with references to the handbooks where the spectra can be located. No actual spectra, but you can get NMR chemical shifts, mass spectral peaks and intensities, and peak values for IR, Raman, and UV/Vis.
[Free] Search for NMR spectra, structures and properties of organic compounds. Also allows for spectrum prediction and submission of NMR shift datasets for peer review.