A bard who takes the Magic Initiate feat and chooses the bard spell list for the feat can use spell slots to cast the 1st-level spell they learn from the feat. A wizard who takes the feat and chooses the wizard list can do the same.
Can an Arcane Trickster rogue or Eldritch Knight fighter who takes the Magic Initiate feat and picks the wizard spell list also use their spell slots to cast the 1st-level spell they pick for the feat?
Both the AT rogue and the EK fighter use the wizard spell list for their Spellcasting feature, so one could assume that they could use their spell slots to cast a wizard spell from the feat... But is that assumption correct?
This Q&A addresses the general case:
Can you cast a spell learned from the Magic Initiate feat using spell slots?
However, the Arcane Trickster rogue and Eldritch Knight fighter are a bit of an exception and aren't specifically addressed by that Q&A. Both the Arcane Trickster rogue and Eldritch Knight fighter use the wizard spell list, even though they are not themselves wizards. As a result, I'm not 100% sure if my assumption about their interaction with the feat is correct.