The correct answer is option 3: 2 and 3 only.

Let`s dissect each statement to understand why this is the correct answer.

Statement 1: "These occur only on gentle slopes during rain."

This statement is incorrect. Landslides can occur on both gentle and steep slopes, and they can be triggered by various factors, not just rain.

Statement 2: "They generally occur in clay-rich soil."

This statement is correct. Landslides are more likely to occur in clay-rich soil because clay has a low permeability, which means it holds water and can become saturated, leading to instability and landslides.

Statement 3: "Earthquakes trigger landslides."

This statement is also correct. Earthquakes can generate strong ground shaking, which can cause slopes to fail and result in landslides. The shaking can loosen the soil and rock, making them more prone to sliding.

Based on our dissection, statement 1 is incorrect, and statement 2 and 3 are correct. Therefore, the correct answer is option 3: 2 and 3 only.