Historically, there were rabbinic authorities who cited the custom of singing this mizmor. Among Polish Jewry, the Matteh Moshe [R. Moshe Mat of Galicia, c. 1591] (see #48) and the Levush [R. Mordecai Yoffe, 1530-1612] (see ch. 51:1;7) indicate such a precedence. This custom was also practiced a generation later in Frankfurt, as indicated by the Yosef Ometz [R. Yosef Yuzpa Han Nordlingen, 1570-1637] (see #277).
It is not common to see this sung nowadays. R. Avraham Yosef (son of R. Ovadia Yosef and chief rabbi of Holon) writes that this is because people are often pressed for time and rush to get to work during the week.
Nevertheless, there have been attempts to sing it again during the week. The most charming attempt, in my opinion, for such a revival comes from the Israeli musician Aaron Razel, who composed and recorded a melody for the mizmor in order to fulfill said halakhah.
As reported in Israeli outlets last year:
בוקר חורפי אחד, הסמטאות מכוסות
בערפילי בוקר קסומים, והנה עוצר אותי חסיד ברסלב 'אהרן, אתה עושה מנגינות
נכון?, אתה מכיר את ההלכה שצריך לשיר את מזמור לתודה? אתה חייב לכתוב
למזור הזה מנגינה כדי שנצא ידי חובה להלכה'. הוא הלך לדרכו, ואני לדרכי
ולא שכחתי את אשר ביקש
וכשהגיע הגיע חודש אדר, פשפש אהרן בערימת הפתקאות והתווים ומחליט להגשים
לחסיד את חלומו. כעת הוא מגיש את "מזמור לתודה", שיר חדש וקופצני המאפשר
לכולנו לקיים את ההלכה ככתבה.
"One winter morning [in Tzfat], when the alleys were covered in a magical morning mist, a Brelov Hasid stops me. He says, 'Aharon, you make melodies, right? You know the halakha that one must sing mizmor l'toda? You have to write a melody for this mizmor so that we can fulfill our halakhic obligation.' He went off his way, and I mine, and I never forgot his request."
When the month of Adar arrived, Aharon searched through his pile of scraps and notes and decided to realize the Hasid's dream. He is now presenting mizmor l'toda, a new and bouncy song that lets us all fulfill the halakha as it is written.
I have heard it sung in Jerusalem, and you can heard the recording here.