To seriously study cycles requires reliable time series data, preferably at regular intervals and for as long a period of time as possible. Data is collected on many different subjects and over many different time frames.
Hyndman Time Series Data Library contains a very wide range of time series data.
Yahoo Finance is an excellent site which has a very wide range of up to date finance data available.
International Stock Market Indices at Yahoo Finance available daily, weekly and monthly going back several decades.
Cross-National Time-Series Data Archive (also referred to as CNTS) offers a comprehensive listing of international and national country data facts. The database has statistical information on a range of countries, with data entries ranging from 1815 to the present.
StatLib---Datasets Archive is datasets from books.
StatLib---JASA Data Archive is datasets from articles published in the Journal of the American Statistical Association.
JISAO - Joint Institute for the Study of Atmosphere and Ocean Gridded Data Sets and Time Series for precipitation, surface temperature, sea-level pressure, surface vector wind & stress and miscellaneous; Links to other weather data and software for analysing data.
Monash - Other Time Series Data is a massive list of sites for getting data.
Codata is not just time series data, but contains all the latest physics, chemistry and fundamental constants.
British, European and Other data archives contains economic, social and demographic data.
UV index & UV dose based on GOME is based on Meteosat cloud cover information
UK Data Archive on almost everything.
WebEc International economics data.
Social Science Data Archives Sources of data from almost every country in the world. Very extensive list of links.
Social Sciences Virtual Library Data archives for much of the world.
Federal Reserve Bank of New York Foreign Exchange XML Data
SuperDARN Data Archives for global convection.
NARA AAD - Access to Archival Databases wide variety of data.
SOPAC includres data and utilities.
Time Series Analysis and its Applications book, free software, documentation and time series data.
California State University Business - Statistics includes business cycle indicators.
University of Maryland EconData Several hundred thousand economic time series, produced by a number U.S. Government agencies and distributed in a variety of formats and media, can be found here. They have been put into a standard, highly efficient, easy-to-use form for personal computers and made publicly available through this site. These series include national income and product accounts (NIPA), labor statistics, price indices, current business indicators, industrial production, information on states and regions, as well as international data.
UK National Statistics The Data Archive holds the largest collection in the UK of computer-readable data on social and economic topics. Data are acquired from academic, commercial and government sources; preserved and made available for further analysis by the research community. The Archive holds over 4,000 datasets. Most relate to post-war Britain, although an increasing number pertaining to earlier periods are now becoming available. The data available extend across the full range social sciences and humanities, and contain information about most areas of social and economic life. In addition to British cross sectoral studies from academic, government and commercial sources, the Archive holds time series data, major longitudinal studies, panel surveys and major cross-sectional studies.
DATA Sources at Dalhousie Canadian and international data.
PhysioBank Archive Index includes ECG, continuous invasive blood pressure, respiration, oxygen saturation, and EEG, among others.
CalCOFI California Cooperative Oceanic Fisheries Investigations The 54+ year hydrographic time-series includes temperature, salinity, oxygen and phosphate observations. In 1961 nutrient analysis expanded to include silicate, nitrate and nitrite; in 1973, chlorophyll was added; in 1984, C14 primary productivity incubations were added.
ESDS International Guide to freely available international data resources.