It depends on the oxidation reagent. Jones reagent is aqueous so maybe dioxane, DMF (not DMSO), maybe THF, acetonitrile.
KMnO4 is also water-soluble so its similar. I think there is tetrabutylamoniumpermanganate also so this could be used in DCM, EtOAc etc. Acetic acid would also be fine in all cases. I have no reference to these suggestions, its just my thoughts.
No, you can use just water but often your startingmaterial is not soluble in just water, you need a co-solvent.