One measure of competition in an industry is via Concentration Ratios. These generally take 2 forms:
In the U.S., these data are collected and measured by the Census Bureau's Economic Census (available every 5 years only), and in recent years made available in the American Factfinder database, organized in tables by NAICS codes.
1) The easiest way to find them is to go to American Factfinder, select Advanced Search, then type "Concentration" in the "topic or table name" box, select Industry Codes in the box and enter your NAICS code and hit Go. This will then display a list of tables with your NAICS industry's concentration ratios (most recent available currently is 2012, until the 2017 survey data is released in 2018).
2) If you'd rather browse the tables at the NAICS sector level, select Advanced Search, then type "Concentration" in the "topic or table name" box, and hit Go.This will then display the Concentration Ratios tables at the NAICS sector level, from which you can drill down to lower level NAICS industries. (you can also search more specifically by NAICS code if you want).
Two federal agencies have primary responsibility for antitrust law enforcement in the U.S. There are any number of entities that have jurisdiction in other countries. You may find economic analyses or legal filings on past or present mergers and acquisitions that shed some light on the competitive nature of individual industries.
Tools used by economists and regulators to measure industry market concentration.
The Census Bureau is one of the best sources for economic & demographic data.
In eStatement Studies you'll find Financial Ratio Benchmark (FRB) figures, broken down by US Region and revenue quartile, for a very large number of industries. Search or browse by NAICS industry codes.