Shulchan Arukh Orach Chayim · Siman 13

English

Having valid Tzitzit on all four corners is an essential requirement: one missing invalidates the whole and the entire four-cornered garment is regarded as not having valid Tzitzit at all. So, someone who goes out (wearing such a garment) into a Public Area on Shabbat would be liable to bring a Sin Offering. (See also Chap. 301 para. 38)

English

If the garment is properly fitted with valid Tzitzit, then it is permitted to go out into a Public Area wearing either a Tallit Katan (small) or a Tallit Gadol (large), even at this time when we do not have the Techeilet (blue-dyed thread). [This permission does not apply if he has simply put the Tallit onto his shoulders]. [We presume the Tallit to be properly fitted with valid Tzitzit, so it is not necessary to check it before going out(into a Public Area on Shabbat)]

English

If, while he is in a Carmelit Area on Shabbat, a person discovers that the Tallit he is wearing is invalid, he should not take it off until he has reached his home, because the respect due to other people is greater. Rema: And even a tallit katan worn under one's clothing does not need to be removed and this is the law if one of the tzitzit has become invalid and one is ashamed to sit without a tallit may wear it without a blessing because of the honor of the public.