Is it possible to mix a drink with a non-standard phase of ice? Would it be possible to safely cool and drink a glass of water with anything else than the Ih form of ice?
Here and here you can see that some alternative forms of ice have a higher density than water, hence they would sink.
Would it be possible to have the coolest party ever, where the attendees would drink from glasses where the ice is sinking instead of floating?
 A: As you can see from the phase diagram plot in the first link you provided, the only other ice phase which is stable at atmospheric pressure is ice XI, and its density is about the same as that of the most familiar ice phase (ice Ih). The other denser ice phases that you see on the phase diagram are only stable at pressures significantly above 1 atmosphere. As far as I'm aware, none of those high-pressure phases of ice are metastable, so you would have no chance of synthesizing any of those high-pressure ice phases in a high-pressure device (e.g., diamond anvil cell, Paris-Edinburgh cell, etc.) and then trying to retain the phase as you download the pressure back to atmospheric pressure.
A: From the diagrams on the webpages you linked, it appears that other ice phases begin to form at around 200 MPa of pressure, and about $-20\text{C}^{\circ}$. Keep in mind normal water freezes at $0\text{C}^{\circ}$, and air pressure at sea level is around 0.101MPa. That means an ice cube made of one of these phases would sublimate or explode very quickly. If it didn't explode, the water in the cup would probably freeze in pretty short order.
