Try 61.89(a)(6)
A student pilot may not act as pilot in command of an aircraft: with a flight or surface visibility of less than 3 statute miles during daylight hours...

Now, that doesn't blanket prohibit special VFR, but your "Student Special VFR" is now limited to 3 miles and clear of clouds.

I think no.
FAR 61.89 (6) indicates a student pilot cannot circumvent the 3 SM regulation (by SVFR)

A student pilot may not do SVFR. Link the regs togeter.

How is a student pilot going to qualify as a private pilot?

Seriously. I'm a student pilot, and if it's below VFR minimums I dont fly, hands down, even if I could I wouldn't. When I get my license I don't think I will put myself in that position. Granted I don't make the weather, but still...In my opinion, if you're requesting SVFR you should have an instrument rating.

That's for night SVFR which was can all agree disqualifies even private pilots without an instrument rating.

On the student pilot issue, the only relevant regulation I know of is the one that requires at least 3 miles visibility and visual contact with the surface. So long as a student pilot can meet those limitations, I see nothing the the regs to prohibit it.

Less that 500' below the clouds with 50 miles visibility to enter Class D airspace and land? Sure. Why not.

FWIW, though, the answer when the question comes up on a CFI FIRC quiz is "no." I've had a discussion with this with Gleim and the theory (other than "the guy at the FSDO who okays the test makes us do it this way") seems to be that unless you are permitted to do everything that you are permitted to do under the clearance, you can't of any of it.

A solo student pilot MAY be given a special VFR clearance, however they still need to have 3SM inflight visibility, so it kind of becomes a moot point... UNLESS they need the SVFR due to cloud bases. (Such as very low overcast).

Simply put, there's no prohibition on a Student Pilot asking for or accepting/being given an SVFR, however all of the applicable rules apply, including 61.89(a)(6):

So the bottom line is:

Your CFI is an idiot for letting you fly in such low weather conditions.

Your judgement is somewhat suspect for flying in them.

You can't do it legally unless you have 3sm vis.

You can always operate under the emergency authority of 91.3, if the sky is falling and you can't go anywhere else.

..although you can still be violated if your own actions caused the emergency.

§61.89 General limitations.

(a) A student pilot may not act as pilot in command of an aircraft:
(1) That is carrying a passenger;
(2) That is carrying property for compensation or hire;
(3) For compensation or hire;
(4) In furtherance of a business;
(5) On an international flight, except that a student pilot may make solo training flights from Haines, Gustavus, or Juneau, Alaska, to White Horse, Yukon, Canada, and return over the province of British Columbia;
(6) With a flight or surface visibility of less than 3 statute miles during daylight hours or 5 statute miles at night;
(7) When the flight cannot be made with visual reference to the surface; or
(8) In a manner contrary to any limitations placed in the pilot's logbook by an authorized instructor.
(b) A student pilot may not act as a required pilot flight crewmember on any aircraft for which more than one pilot is required by the type certificate of the aircraft or regulations under which the flight is conducted, except when receiving flight training from an authorized instructor on board an airship, and no person other than a required flight crewmember is carried on the aircraft.
(c) A student pilot seeking a sport pilot certificate must comply with the provisions of paragraphs (a) and (b) of this section and may not act as pilot in command—
(1) Of an aircraft other than a light-sport aircraft;
(2) At night;
(3) At an altitude of more than 10,000 feet MSL or 2,000 feet AGL, whichever is higher;
(4) In Class B, C, and D airspace, at an airport located in Class B, C, or D airspace, and to, from, through, or on an airport having an operational control tower without having received the ground and flight training specified in §61.94 and an endorsement from an authorized instructor;
(5) Of a light-sport aircraft without having received the applicable ground training, flight training, and instructor endorsements specified in §61.327 (a) and (b).