What script is Undercommon written in? I think it's supposed to be the common script, but I'm just confirming that.
The PHB says "Elvish" on p123. There isn't any information in the Sword Coast Adventurer's Guide, where the other styles are specifically named, though "Elvish" clearly maps to "Espruar", and "Dwarvish" to "Dethek".
The Forgotten Realms Campaign Setting, a 3E book, specifically refers to Espruar, which is the proper name of the script used by Dambrathan and Elvish, among other languages. Earlier works (like the 1E FRCS) indicate Espruar was specifically/originally used by silver elves.
In the language section of the PHB, they show Elvish, Dwarvish, and Draconic character equivalents to English letters. This implies that there are letter commonalities across all the languages, even if the specific words and letter frequencies vary between languages.
Considering where most people who speak Undercommon live, it's likely that some Undercommon is also written with Dethek lettering.
The Dwarvish language of Faerun uses a runic alphabet called Dethek, whose characters are easy to etch into stone and metal, as evidenced by the runestones and way-markers found in ancient dwarven tunnels and mines. - SCAG, p103.