I've always felt that when it comes to rapid transit (i.e. grade-separated, electrified), interlining is a great way to double frequencies along shared route segments (two tracks) and is an easier, quicker, and more cost-effective approach to expanding rapid transit systems (as opposed to building single lines from scratch) because of the existing ridership base.
The downside of interlining includes reduction in the quality of service, with trains not being able to run as frequently, schedule delays more likely, and disruptions caused by construction.
In LA, the best case for interlining involves a potential heavy rail extension of the B (Red) Line down Vermont from Wilshire to 120th Street (and eventually farther south). Vermont has the second-highest bus ridership after Wilshire, so I see value in trains arriving every 2.5 minutes along the shared spine and 5 minutes along the branches during peak periods.
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