FAR 91.157(a) uses the phrase “within the airspace contained by the upward extension of the lateral boundaries of the controlled airspace designated to the surface for an airport” to indicate where a SVFR clearance may be authorized.
The FAA often uses the phrase "surface area" to refer to the controlled airspace that actually surrounds the airport for which it is designated, specifically excluding any surface-level Class E "extensions". An argument can be made that in the context of interpreting the meaning of the FARs, this is the most correct understanding of the phrase "surface area". This interpretation is based on the difference between the language used to describe surface-level Class E "extensions", and the language used to describe other surface-level Class E airspace, in the "Airspace Designations and Reporting Points" document, current edition FAA Order JO 7400.11E. For more on this, see the related ASE question What indication has the FAA given that phrases like "surface area of Class E airspace designated for an airport" do or don't include E4 "extensions"?.
But the phrase "surface area" does not appear in FAR 91.157(a).
However, the phrase "designated for an airport" does appear in FAR 91.157(a). An argument can be made -- again, based on based on the difference between the language used to describe surface-level Class E "extensions", and the language used to describe other surface-level Class E airspace, in the "Airspace Designations and Reporting Points" document, current edition FAA Order JO 7400.11E -- that this phrase excludes surface-level Class E "extensions". Again, for more on this, see the related ASE question What indication has the FAA given that phrases like "surface area of Class E airspace designated for an airport" do or don't include E4 "extensions"?. According to this interpretation, FAR 91.157(a) would not include the surface-level Class E "extensions", so Special VFR should not be authorized there.
However, one problem with this argument is that FAR 71.71(b) seems to either 1) use the phrase "designated for an airport" in its description of all surface-level Class E airspace, or 2) omit any reference to surface-level Class E "extensions" at all, in which case it seems the "extensions" might be construed not to legally exist at all. Obviously, the latter interpretation is somewhat problematic. This seems to imply that the phrase "designated for an airport" should not be construed to exclude surface-level Class E "extensions".
It is also unclear as to whether or not whether FAR 91.155(c), the regulation against flying below a ceiling of 1000' AGL or lower, applies in the surface-level Class E "extensions", because the language of FAR 91.155(c) is very parallel to the language of FAR 91.157(a). Again, the phrase "designated for an airport" is used, but the phrase "surface area" is not. For more on this, see the related ASE answer "Does FAR 91.155c apply to class E surface extensions?"
The last paragraph of this 2010 response from the FAA ATO Western Service Center concludes that due to the wording of FAR 91.157(a), SVFR should not be authorized in surface-level Class E "extensions".
One problem with the 2010 response from the FAA ATO Western Service Center is that is contains the passage "The language in FAR 91.157 exactly matches the language in FAAO 7400.9T par 6002.". This is not actually quite true, because FAAO 7400.9T par 6002. contains the phrase "The Class E airspace areas listed below are designated as a surface area for an airport." As we've already noted above, FAR 91.157(a) does not actually contain the phrase "surface area".
Naturally, for specific airspaces, sometimes there are also specific operational considerations that preclude authorization of Special VFR flight in surface-level Class E "extensions". One such case is noted in the second-to-last paragraph of this 2010 response from the FAA ATO Western Service Center (same link as immediately above). However, the related ASE answer A: In the US, in actual practice, workload permitting, will ARTC facilities grant SVFR clearance for surface-level Class E “extensions” (E4 airspace)? gives examples of similar situations where controllers are willing to authorize Special VFR flight in surface-level Class E "extensions".
FAA Order 7110.65Y, entitled "Air Traffic Control", states in section 7-5-1 that Special VFR flight will only be authorized "within the lateral boundaries of Class B, Class C, Class D, or Class E surface areas, below 10,000 feet MSL." As noted above, there are many instances where the FAA has used the phrase "surface area" to refer only to the controlled airspace that actually surrounds the airport for which it is designated, specifically excluding any surface-level Class E "extensions". So this passage could be construed as guidance that Special VFR flight should not be authorized in surface-level Class E "extensions". But it's not clear that this guidance would really be congruent with the best interpretation of the meaning of FAR 91.157(a).
In actual practice, controllers will often authorize SVFR flight in surface-level Class E "extensions". For more on this, see this related ASE answer. (And controllers generally use the word "surface area" in its most logical sense-- meaning the entire column of controlled airspace that is in contact with the earth's surface, including any surface-level Class E "extensions".)
The truth is that the FAA is deeply divided as to whether phrases in the FARs like "surface area", "designated for an airport", and "surface area of controlled airspace designated for an airport" should or should not be construed to include surface-level Class E "extensions".1 Ambiguity around this issue has existed ever since the 1993 "alphabet" airspace re-designation.2 There's some consideration of a future rule-making effort to address this issue.3 It's possible that the language in AIM 3-2-6, as well as in the legend of the Chart Supplements, that reads "Surface area arrival extensions become part of the surface area and are in effect during the same times as the surface area" will be changed in the future to avoid implying that all regulations that apply to the core "surface area" should automatically be construed to apply to the surface-level Class E "extensions" as well.4
Due the confusion around this issue, whenever a pilot is issued a SVFR clearance for a given controlled airspace, he or she should verify whether or not it includes any adjoining "extensions", in any case where that might be a factor.
Footnotes:
1,2,3,4) Source-- comments made by high-level FAA staffers during April 2021 airspace charting meeting.
Related ASE questions and answers
A: Does an SVFR clearance extend to Echo surface extensions?
A: In the US, in actual practice, workload permitting, will ARTC facilities grant SVFR clearance for surface-level Class E “extensions” (E4 airspace)?
A: Does FAR 91.155c apply to class E surface extensions?
Q: What indication has the FAA given that phrases like "surface area of Class E airspace designated for an airport" do or don't include E4 "extensions"?
A: What indication has the FAA given that phrases like “surface area of Class E airspace designated for an airport” do or don't include E4 “extensions”?
A: Is the official weather report for an airport in Class D airspace controlling for its Class E surface extension?
Q: Which parts of class E airspace can an ultralight (part 103) fly in without prior ATC authorization?