If a sorcerer wild mage rolls on the wild magic table and ends up casting a fireball centered on themself, can the mage use Careful Spell when they cast that fireball, allowing the mage and some of their party members to succeed on their saving throws against the spell and take half damage?
The newly released errata for the PHB answers this question definitively:
If a Wild Magic effect is a spell, it’s too wild to be affected by Metamagic. If it normally requires concentration, it doesn’t require concentration in this case; the spell lasts for its full duration.
So you can't use any form of metamagic on spells cast by your Wild Magic.
The wording on meta magic is "when you cast" and the wording on the wild mage table is "you cast." Since you are casting a spell, and have access to meta magic, you can definitely us meta magic such as careful spell on the spells you cast via the wild magic table.
Additionally, there is nothing in the Wild Magic sorcerer writeup that indicates that meta magic cannot be used.