A bond that pays interest forever and has no maturity is a perpetual bond. In what respect is a perpetual bond similar to a no-growth common stock? Are there preferred stocks that are evaluated similarly to perpetual bonds and other preferred stocks that are more like bonds with finite lives?

First and foremost just to let you know, perpetuity is nonexistent at these days. Last I remember the British swapped their perpetuity to a 100 year finite bond.

I don't know what is no growth common stock and I'm not even sure if there is anything such as that.

You can think a preferred stock as a perpetual bond assuming the company will outlive your life and will make dividend payments. Dividend payments are of course no guarantees as the company can halt making dividend payments anytime they prefer. It also matters what kind of preferred you own. Definition wise preferred is not anything like a bond because a bond is obviously a debt obligation and the preferred is owner's equity. Debt is applied in income statement and preferred dividends aren't. Debt comes in the balance before preferred.

I hope this helps.