Generally speaking, the answer is "no" for everyday situations like apartment-rental and job applications -- but there are a few exceptions that vary by state.
Here are five situations that may require you to disclose an expungement, depending on where you live:
Applying for a state-issued professional license. If you're pursuing a license to practice law in any state, you are required to disclose your expungements. Other state-licensed professions may also fall under this exception, so it may be wise to consult a local criminal defense attorney who's familiar with the laws of your state.
Applying for a law-enforcement job. Disclosure is required for a federal law-enforcement position, such as with the FBI. In some states, you must also disclose expungements if you want to be a local police officer, or even work as support staff with the agency.
Applying for a school-related job. Some states require expungement disclosure for any job at a school, public or private. Other jobs that involve close contact with children may also require disclosure.
Buying a firearm, or applying for a concealed carry permit. This may seem logical, but not every state requires it. If it's required where you live, you'll see it on the permit application.
Applying for public office. You may not be a crook, but you will likely be required to disclose all expungements if you choose to run for any level of elected office, such as governor, city council member, or even dog catcher (if that's an elected office where you live).
Now, keep in mind that in many cases, disclosing an expungement does not necessarily mean you won't get hired. (In some states like Maryland, failure to disclose an expungement can't even be the sole reason for refusing to hire a candidate.) Because the laws about expungement are complicated, it's best to consult a local attorney with any questions about your particular situation.