Which adhesive is best for vinyl wall panels?

For vinyl wall panels, you definitely need an adhesive that's been specifically developed for this substrate. Silicone adhesive works best here, or even better a polyurethane adhesive, because they're flexible enough and move along with the vinyl - that's important since vinyl shrinks and expands with temperature changes. Regular super glues or wood adhesives aren't really suitable because they're too rigid and the panels can then peel off. Many people also use special PVC/vinyl adhesive, which you can find in any hardware store.

Really important: check the packaging to make sure the adhesive is explicitly approved for vinyl. Before you glue it on, you need to thoroughly clean and dry the wall surface - dust and grease are the enemy. For larger areas, it's better to use area adhesive rather than spot adhesive, so the panels bond everywhere and don't sag later. Also let the adhesive dry properly after applying it before you put any stress on the wall - with most products that takes between 24 and 48 hours.