Flat extension cords are fine under rugs — but only if they're rated for it.
Yes, flat cords are actually built for this, but the important difference isn’t shape — it’s whether they’re listed for under-carpet use.
Most round extension cords are not rated to go under rugs or carpet. They’re thicker, trap heat, and are a fire hazard if covered. Flat cords are thinner, dissipate heat better, and many are specifically designed for this. Look for “carpet” or “under rug” on the package.
That said, flat cords also tend to carry less current (typically 13 amps vs 15 amps for a standard round cord). So don’t run a space heater through one under your rug. That’s a bad idea regardless.
For baseboard use: either works. Flat is easier to tuck away, but round is more durable for long-term, visible runs.
Under-carpet? Only flat, and only if listed for it. Don’t try to save ten bucks by using a round cord you already own.