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VFR minima are outlined in 91.155.
There is no hard and fast 1000 & 3 rule.
Depends on controlled vs. uncontrolled airspace, 91.155 (c) requires a 1000 foot ceiling if operating below the ceiling in controlled airspace which goes to the surface (class B, C, D & E surface area).
Outside these boundaries, in controlled airspace, you need to be 500 feet below the clouds.  Couple that with minimum altitude requirements in 91.119 (500 feet AGL over uncongested areas) and you get a minimum 1000 foot ceiling.
In uncontrolled airspace (say below a transition area with a 700′ AGL base), during the day, you could be at 699′ AGL below a 700′ ceiling with 1 mile visibility and be legal for VFR flight.

(3-152) = (3 miles 1000 above 500 below and 2000 horizontal)

Special VFR

Per 91.157, special VFR ops are only allowed when flight visibility is at least 1SM, but to takeoff or land under special VFR you need reported ground visibility of 1SM. If ground visibility isn't reported, then flight visibility may be substituted under certain circumstances (a little complicated, see the reg).