In the state of Utah, though uncommon, it is possible to get a DUI while riding a bicycle. Typically, if you are ticketed for being intoxicated while riding a bicycle you will be charged with public intoxication rather than a DUI but in some circumstances, people have received DUIs. Often these cases can be argued because the law states that a DUI is for driving a "vehicle" under the influence and there is some wiggle room in proving whether or not a bicycle should be considered a vehicle.
The definition of a vehicle according to the law is "a device in, on, or by which a person or property is or may be transported or drawn on a highway." Technicalities such as what constitutes a highway, or what is defined as a vehicle are often brought to the table when arguing such cases. The bottom line is that judges are able to interpret the law and definitions as they see fit. If they deem your actions to have the potential to cause extreme harm to yourself or others, the charge may stick.
The long term implications of this law are the potential for other methods of transportation to be included as vehicles such as roller blades, scooters, and skateboards. There was even a case where two men on horseback were convicted of drunk riding, though the case was eventually overturned.
In the short term, if you are planning on drinking more alcohol than the state's limit (which was recently lowered to 0.05%) be sure to have a sober driver with you or take a taxi, Lift or Uber home because riding your bike may not be your safest option.