Teacher Paul and Michael *Sigma* are right: it's "Today is my favorite fashion icon's birthday".
We use the apostrophe to spell the possessive form of a name, of a noun, or of a word that functions as a noun.
The rules are different for nouns and names.
1. For singular nouns, always add 's: boy -> boy's; goose -> goose's; lass -> lass's; woman -> woman's; child -> child's
2.a. For plural nouns that end in -s, add only '; boys -> boys'; lasses -> lasses'
2.b. For plural nouns that do not end in -s, add 's: geese -> geese's; women -> women's; children -> children's
For names that do not end in the sound of "s", add 's: John -> John's; Mary -> Mary's; Smith -> Smith's
For names that end in the sound of "s", there are three ways.
1. Some native speakers always add the syllable "iz" to the end of the name, and they always add 's: Jesus -> Jesus's; Moses -> Moses's; Jones -> Jones's
2. Some native speakers never add the syllable "iz" to the end of the name, and they always add only ': Jesus -> Jesus'; Moses -> Moses'; Jones -> Jones'
3. Some native speakers add the syllable "iz" to some names, but do not add the syllable "iz" to other names. Those people usually write 's after the names to which they add the syllable "iz", and they usually write only ' after the names to which they do not add the syllable "iz": Jesus -> Jesus'; Moses -> Moses'; but Jones -> Jones's.
The conventions for some compound nouns seem strange to some people. For example
Singular: sister-in-law -> sister-in-law's
court-martial -> court-martial's
passer-by -> passer-by's
Plural: sisters-in-law -> sisters-in-law's
courts-martial OR court-martials -> courts-martial's OR court-martials'
passers-by -> passers-by's