Contrary to popular belief both illegal and legal immigrants are far less likely to commit crimes.
This report from the Cato Institute (with statistics from the Texas Department of Public Safety (DPS)) shows that immigrants are far less likely to commit crimes than the native born population.
As this chart shows, the total amount of convictions by illegal immigrants are less than half of the rate of the native born population. The amount of convictions for legal immigrants is less than 1/6 that of the native born population.
Homicide rates (a good proxy for violent crimes) are substantially lower among immigrant groups as well.
Rates of petty crime, like larceny are virtually unheard of when it comes to immigrants. Illegal immigrants commit larceny 80% less than native born citizens
"Notably, we found a significantly lower rate of violence among Mexican-Americans compared to blacks and whites . . . In particular, first-generation immigrants (those born outside the United States) were 45 percent less likely to commit violence than third-generation Americans." - Sampson(2008)
"In a large-scale collaboration by four universities, led by Robert Adelman, a sociologist at the State University of New York at Buffalo, researchers compared immigration rates with crime rates for 200 metropolitan areas over the last several decades.... According to data from the study, a large majority of the areas have many more immigrants today than they did in 1980 and fewer violent crimes." - New York Times
"Empirical studies of immigrant criminality generally find that immigrants do not increase local crime rates and are less likely to cause crime than their native-born peers, and that natives are more likely to be incarcerated than immigrants."
- Cato Institute