Is it permissible to serve a birthday cake (no candles) on Shabbos morning/mid-day Kiddush in
(a) Shul/Public gathering
(b) One’s own house
(1) The first 2 Shabbasos of the 3 Weeks
(2) ‘Regular’ Shabbos Chazon
(3) When Shabbos Chazon is 8 Av
(4) When Shabbos Chazon is 9 Av
It is always permitted to serve a birthday cake on Shabbos.
However, if the cake will not be missed – it is not “expected” by anybody – it is correct to refrain from serving it on Shabbos that falls on the 9th of Av.
The reason for this is that the common custom is not to engage in any form of avelus on Shabbos Chazon, including wearing Shabbos clothes (see Rema 551:1; Mishnah Berurah 551:6). It is the more so permitted to serve a birthday cake, which is not in any way a contravention of laws of mourning.
Yet, for Tisha Be’Av itself, one must mourn for matters that are “private.” Strictly speaking this refers to marital relations alone, and not to other matters. Still, a “birthday celebration” is not in the spirit of the day, and is better to avoid.