what kind of front produces gentle rain showers

A warm front is the type of front that produces gentle rain showers.

Why a warm front gives gentle rain

When a warm front moves into an area, warm, moist air slides slowly up and over a cooler, denser air mass.

Because this lifting is gradual, it forms layered clouds (often stratus) and leads to light, steady precipitation rather than short, violent storms.

Quick example

Imagine mild, moist air from over the ocean moving inland over a region of cooler air.
As that warm air is gently pushed upward along the front, clouds thicken and you get hours of light, gentle rain instead of intense downpours.

TL;DR: The front that produces gentle rain showers is a warm front.

Information gathered from public forums or data available on the internet and portrayed here.