First rule of electrical engineering, especially when using live power.
DO NOT ELECTROCUTE ANYONE!
When using live power, including the ground is always your best solution. For the extra cost of cable and plugs it buys you so much more peace of mind.
If grounding is completely off the table then you need to double insulate to make sure the operator can never touch a metal part. HOWEVER. This can ONLY be applied if your widget has no external connections other than power. You can not have a DB9 serial port sticking out the side of the plastic.
As for shielding within a double insulated design. You can in fact connect the casing to the neutral line for EMI reasons. The neutral is normally attached to ground somewhere, either at the breaker box or at the line transformer.
However, if you do the latter, be aware that if you open up the box there is a chance the outlet is wired backwards and the frame will be live.
As such you need that ubiquitous "DO NOT OPEN CASE. NO USER SERVICABLE PARTS" sign on the outside of the casing.
If you are using a metal enclosure with mains coming in with no ground you will not be able to attain a safety certificate to allow you to sell this device pretty much anywhere on the planet.
There is one exception to that where you can prove that the case is double insulated from the contents of the box. Double Insulated (Class II) Appliances. Even then, the certifier will frown rather a lot and be reluctant.
Further, in that scenario there can be no un-isolated electrical connection to anything inside the box, so connecting it to the 5V return would be a no-no.