I assume you are asking about US regulations;
The link is here:
http://ecfr.gpoaccess.gov/cgi/t/text...2.5.34&idno=14
As usual some interpretation is required, therefore this is my interpretation.
You are only allowed to take off under special VFR if the airport is towered or a satellite airport to a Class D,C or Bravo airspace.
Most non towered airports have overlying class E that goes down to 700', on the VFR chart( sectional) marked as a magenta line like this one;
Airports that have the lower (700') E above them have instrument approaches. The lower E is meant to protect the instrument approaches, similar to airspace extensions. Nobody is supposed to be flying in the traffic pattern or elsewhere VFR because in E the VFR minima are 3 mi 500' below 100' above and 200' away from cloud.
Departing with a 500' overcast would put you in class G airspace where no clearance is required and you are legally allowed to fly with 1 mile visibility and clear of clouds.
Remember that a lot of non precision approaches have 1 mile vis 500' as an approach minimum. For all intends and purposes that is IFR.
Special VFR is meant to depart an airport with marginal VFR conditions towards an area with better weather close by.
Or to land at towered airport when conditions are temporarily marginal VFR and you need some extra help.