What's the construction of the island?
If it's wood, I'd be tempted to use a pin nailer and possibly even forego adhesive. The pin nails are surprisingly tenacious and very nearly invisible.
Titebond III (or other simialar wood glue) is very strong, but requires pretty decent clamping while drying to work to its best. It's also not too nasty to work with; it cleans up nicely with a damp rag and doesn't tend to turn you into a human post-it note. Practically all modern wood glues are stronger than wood when the joints are fitted and clamped properly. Titebond III is also nice because it is waterproof, or at least as waterproof as wood is, and is approved for indirect food contact (i.e. you can use it to glue up a cutting board).
Thickened epoxy is also very strong and requires less pressure when clamping, but is messier to use, can cause allergic sensitization (so should be used with gloves), and can be nearly impossible to remove if it drips and cures where you don't want it.
I have used some construction adhesive with very good effect—I think it was PL premium. Very strong when dry, and it adhered to aluminum, wood, and styrofoam (and probably most everything else).
Needless to say, a key with most any glue (other than contact cement, I guess) is to properly brace/clamp things while the glue cures.