The opening mishna of the fourth perek of Megilah teaches that when an item with kedusha is sold, one must use the money only to buy an item of greater kedusha. In its first example, the mishnah says that if a community sells the city street they should use the money to buy a shul, which has even greater kedusha. The Gemara (26a) explains that the street has kedusha because people would daven outside, in the street, on fast days and ma’amadot, the days when it was the city’s turn to represent Bnei Yisrael in the daily karbonot. Furthermore, the gemara explains that this is only the opinion of Rebbi Menachem bar Yosi; however, the Rabanan, whom we pasken like, hold that since the street is only used occasionally and on an irregular basis, it has no kedusha.
Rav Yair Bachrach (Germany, 1638-1702, Shut Chavot Yair 59) discusses the status of a room in a house used as a shul for only a few times a year, but on set dates. Should we view this as a beit knesset with kedusha since it is used on set dates, or should we say that is similar to the city street since it is used so infrequently and it therefore has no kedusha? He points out that this might depend on the different versions of our gemara. Our version of the gemara (and the version of the Rif and Rambam) says that people davened in the street on the ma’amadot which were set dates a few times each year. Nevertheless, the Rabanan say there is no kedusha in the street because it is used so infrequently, even though it is used on set dates. However, Tosfot (25b) say that this version of the text is incorrect, and the word “ma’amadot” should be erased from the gemara. (See Rashi for a more detailed discussion of these two versions.) According to Tosfot, the only time they davened in the street was on fast days for troubles like lack of rain. These fast days were both uncommon and had no set schedule. The Rabanan say there is no kedusha in such streets, but perhaps they would agree that a place used infrequently, but on a set schedule, does have kedusha. The Chavot Yair accepts Tosfot’s version, and paskens that places used as a shul on set times, even if infrequent, have kedushat beit knesset (on those set dates), and therefore must be treated with the proper respect and laws of a beit knesset.
The Poskim also discuss rooms or courtyards right next to a shul that are used by the occasional overflow crowd from the main sanctuary or by people who come late. Rav Yosef Trani (Safed and Constantinople, 1568-1639, Shut Maharit 2:4) says that this is similar to the city streets that our gemara discusses. Therefore, if the room is only used occasionally, it would not have kedusha; however, if it is used regularly by people who come late, it would have kedusha. Rav Shmuel Engel (Poland, 1853-1935, Shut Maharash Engel 1:63), though, writes that the reason the city street in our gemara doesn’t have kedusha is not only because it was used on an irregular schedule for davening, but also because it had other permanent uses as a street and marketplace. However, a room which has no other permanent use, even if it is only used for davening sporadically, has kedushat beit knesset. The Mishnah Berurah (154:3) quotes and paskens like the Maharit that the rooms off of a shul only have kedusha if they are used for davening on a regular basis, seemingly even if they have no other use.