Watch out with CUI. In some states it's the same crime as driving a motor vehicle while under the influence.
In my draconian state, for example, the law prohibits operating a "vehicle", as opposed to a "motor vehicle", while under the influence, _or_ with alcohol to blood/breath/urine ratios above a certain amount. (These are two different offenses. Get caught drinking while operating a vehicle and you'll be charged with two crimes. A lot of other states are like this, too.) Here, bikes aren't vehicles, but riders have all of the "rights and responsibilities" of operators of vehicles. The courts have settled the specific issue of CUI, too-- mounted cyclists are criminally liable just as motor vehicle drivers are for operating under the influence. I know for sure cyclists here have been convicted under the DUI statute.
I strongly suspect that cyclists can proudly count themselves on a par with drivers of vehicles in another important respect, too: our driver's licenses are contingent on our "consent" to our choice of blood/urine/breath tests should the constabulary decide we seem intoxicated. In other words, refuse to take these tests, even on a _bicycle_, and your driver's license could be forfeit for a year. (This, of course, is independent of whatever penalties you might suffer should you end up being convicted of DUI anyway.) I don't know of case in which a cyclist lost his license for refusing these requested tests, but from the precendents in other CUI cases that's what I think would happen.
Of course, losing a license is not quite the same punishment for a cyclist as for a motorist. Interestingly, even the sentencing schedule for DUI calls only for longer and longer license suspensions, as opposed to prohibitions on vehicle operation on public roads. So, at least in principle, even multiple convictions for CUI wouldn't affect your ability to cycle on public roads. Convictions would, though, land you in the joint as well as costing you an arm and a leg. And a judge might order you to stop riding anyway.
If you really want to get drunk, what you ought to do in my state is ride a horse, or a horse-drawn carriage. You can be passed out drunk on these conveyances and there is no per se crime. Yes, you could be charged with being drunk in public, or various traffic offenses, but these are a much smaller deal than driving drunk.
But, anyway, my guess is that you enjoy one practical advantage cycling drunk instead of driving. Cops are less likely to arrest you. You've probably got a decent chance of just receiving an order to walk home. This, too, is just a guess, and in my own case just a theoretical concern anyway, but there it is.
The last state I lived in also didn't distinguish between cyclists and motorists with respect to DUI. Yet, I was never asked to stop at the DUI checkpoints I would sometimes encounter. The cops just ignored me. I don't think the troopers even knew it was a crime to cycle while drunk. (Of course, if I _had_ been stopped, I would just have turned around the next time I saw a roadblock and taken another route home, or just walked off of the street for a ways and then remounted. Another advantage of cycling as opposed to driving...) I would have pretty ticked off if they did stop me, but what could I have done but comply?
Which makes me curious: anyone here been _stopped_ at a DUI checkpoint on your rig?
Also, while I'm on the subject, I do know of a case a few years ago in Washington State in which a cyclist appealed a CUI conviction all the way the State Supreme Court. He won. In Washington, the relevant statutes contain "motor vehicle" at crucial points, though they just refer to "vehicles" at other points. The S.C. ruled, correctly, that this ambiguity in the law had to be resolved in the way favorable to the defendant. So, as far as I know, it's not a crime to cycle while drunk in Washington. At least, that's how it was a few years ago. Perhaps the legislature acted quickly to close this "loophole" after the case was resolved.