Is there any specific prohibition that forbids showing affection with one's spouse in public? For example kissing on the forehead or holding hands, not more extreme affection actions of course.
The Rama writes (EH 21:5)
יש אומרים דאין לנהוג אפילו עם אשתו בדברים של חיבה, כגון לעיין ברישיה אם יש לו כינים, בפני אחרים
Some say that you shouldn't act with your wife in affectionate matters, such as her checking his head for lice, in front of others.
While some have extended this to prohibit any public indication of her Niddah status (such as via non-affectionate touching) as a formal prohibition, Rav Yehuda Herzl Henkin (Benei Vanim 1:37:1) quotes the source of the Rama's citation (the Nimukei Yosef) who simply states:
דרך ארץ שאינו ראוי להתנהג עם אשתו כיוצא בדברים אלה בפני אחרים
It is derekh eretz to not act with his wife in matters such as [her checking his hair for lice] in front of others.
Thus, no formal prohibition is involved. R Henkin writes further that even for a Talmid Chacham who is accustomed to prishut such as this, holding hands in public is ok as it is not particularly "affectionate", unlike lice checking was in those days where you would lie in the checker's arms on a bed. Indeed classically, he notes, women wore special clothing to make it obvious to all when they were Niddot.
Obviously, every circumstance and action is different, so consult your rabbi for a particular ruling.