Does my phone support eSIM?
One check works on any phone, and takes ten seconds.
Dial *#06#
Open your phone app and dial *#06#. If the screen shows a number labelled EID, your phone supports eSIM.
Is your phone unlocked?
A phone bought on a contract can be locked to one carrier, and then no other eSIM will connect. On iPhone, open Settings, General, About and look for Carrier Lock: it should say No SIM restrictions. On Android, ask your carrier, or try another carrier's SIM.
Common phones with eSIM
iPhone
iPhone XS, XS Max and XR, and every iPhone since, including the SE (2nd and 3rd generation). Most iPhones sold in mainland China use two physical SIM slots instead.
Google Pixel
Pixel 3 and every Pixel since. A few early models sold in some countries or by some carriers left eSIM out, so run the *#06# check on a Pixel 3 or 3a.
Samsung Galaxy
Galaxy S20 and later S series phones, the Galaxy Z Fold and Z Flip series, and the Note 20. Many recent A series phones too. Samsung leaves eSIM out of some models in some countries, so run the *#06# check to be sure.
Other brands
Many recent phones from Motorola, OnePlus, Xiaomi, Oppo and Sony have eSIM, but it depends on the exact model and the country it was sold in. The *#06# check is the quickest answer.