As far as I understand, most, if not all of the oils on the shelf at the local auto parts stores do not contain zinc anymore. I am not sure of the reasoning, but I have read this.
Street car oils at the local auto parts store DO indeed contain zinc but have it in lower quantities than in past years.
If I remember the articles I read a few years ago correctly, Zinc (ZDDP levels in general) have been decreasing due to tighter exhaust emissions standards. "High" levels of ZDDP have also shown to decrease the life and performance of catalytic converters. Google ZDDP for more discussions on the topic.
Street car oils typically have around 800 ppm of ZDDP on average while race oils have closer to 1800 ppm or more depending on the other additives the oil manufacturer uses. More than street levels of ZDDP is good for racing environment but levels of ZDDP isn't the only attribute that makes a good oil. There is such a thing as too much...