U.S. nationals, including U.S. dual nationals, must use a U.S. passport to enter and leave the United States. U.S. dual nationals may also be required by the country of their foreign nationality to use that country's passport to enter and leave that country.
Singapore has taken first place on the latest Henley Passport Index 2023 rankings. Singaporeans enjoy visa-free access to 192 travel destinations out of 227 worldwide. With Japan falling to third place, three European countries tie in second place: Germany, Italy and Spain with visa-free access to 190 destinations.
You can only remain outside the United States, however, as long as you re-enter the U.S. before the expiration date printed on your travel document. The travel document is valid for one year after it's issued, typically within 150 days (in some cases longer) after submitting your application materials to USCIS.
In most countries, you can travel anywhere within your nation's borders without a passport. The only exception is when you leave your home country to enter a foreign nation. However, even if you travel nationally, you will always need photo ID, such as a driver's license or National ID card to board a flight.
There are no penalties. The customs officials might be furious but the only thing they can do is confirm you're citizen and let you back in. As for leaving the US, the US doesn't have exit checkpoints so you can leave on any passport.