A sidur I used this morning had the introductory line of the shir shel yom as "hayom yom sheni b'shabas". When I checked the two sidurim I trust the most for excatitude in these matters, one had "bashabas" and the other omitted the line completely. Which is the correct version?
Grammatically speaking, the difference between b'-shabbos and ba-shabbos is that the the former contains just the "ב" prefix, while the latter is a contraction of both the "ב" and "ה" prefixes. So one would translate "b'-shabbos" as "of Shabbos" and "ba-shabbos" as "of the Shabbos".
That said, it would depend on what the intention is of the phrase "היום יום __ בשבת".
Simply speaking, one could say that "שבת" in this context means "week". Indeed, the Metsudah siddur translates, "Today is the __ day of the week." In this case, we need the "ה" prefix for "the week", so the word becomes "ba-shabbos".
On the other hand, I found a Breslov siddur in my house that has a commentary with (typically) kabbalistic tendencies. In it, it quotes from Likutei Moharan (2:2) that everyday one should mention the Shabbos thereby extending its holiness through the entire week, and each day will have its own aspect of holiness, which is why we have a particular mizmor for each day.
According to this, it would seem that the phrase "היום יום __ בשבת" really means "Today is the __ day of [the extended holiness of] Shabbos", in which case there is no need for the "ה" prefix, and the word will be "b'-shabbos". (The Breslov siddur indeed has "b'-shabbos".)
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