They are much more stable than other locations but are not completely stable, so station keeping is required. In a theoretical two body system Lagrange points would be perfectly stable but that is not the case with the solar system. Eg: The orbit of the moon ever so slightly effects the Sun - Earth Lagrange points.

The JWST is a good example. The expected observational lifespan of the telescope is based on how long it is able to remain at L2.

NASA Says Webb’s Excess Fuel Likely to Extend its Lifetime Expectations