Why Do Some Sinks Have 3 Holes?
If your bathroom sink has three holes at the top near the rim, you can safely stop the drain and forget to turn off the faucet, and you probably won't be greeted by a minor flood when you return to the bathroom. Those three holes are overflow holes, and preventing floods is exactly why they are there.
Three holes are something of a design rarity. A single circular or oval hole is much more common, and it can be located under the faucet or on the opposite side of the sink. The overflow opening, whether it's a single hole or three, connects to a channel that leads to the drain. Because the channel has to be molded into the sink, porcelain sinks, with their pros and cons, are more likely to have one than stainless steel or copper sinks. Bathroom sinks usually have overflow openings because they are small and shallow, but kitchen sinks, which are deeper and hold more water, seldom need one.
The way the overflow opening works is simple: water enters the opening, flows through the channel built into the sink, and empties into the drain. In order for this to happen, the drain flange has to have an opening that aligns with the channel. If you're replacing the drain on a sink with overflow holes, correct bathroom sink plumbing demands you buy a drain assembly with an overflow opening, or the overflow holes will be useless.
Overflow holes can also prevent drain clogs
There's another, less obvious function of overflow holes. These small openings allow air into the drain to help keep water moving. In short, they vent the drain. Bathroom sink drains are not very wide, and it's easy for water in a narrow drain to back up before it reaches the P-trap and the waste system. The result is gurgling at the drain opening, possible backflow from the drain to the sink, and an increased likelihood of clogs and standing water. Sink overflow holes help prevent this.
Every sink in a house is already connected to a vent network, as per plumbing codes. The connection is made on the downstream side of the P-trap to keep water flowing from there to the sewer. That connection, however, doesn't directly affect flow through the sink drain and tailpiece, which are separated from the waste line by the water in the trap. The overflow holes allow air into this crucial part of the drain, and because water can flow freely, clogs caused by hair and soap scum are less likely to happen.
Overflow holes aren't standard equipment on all sinks, partly because kitchen and laundry sinks are so large that they don't need them, and partly because overflow channels can get dirty and moldy. If your sink has overflow holes, it's a good idea to clean them periodically by pouring boiling water into them with a funnel. A combination of baking soda and vinegar can also provide effective cleaning.