How can one determine if a Service Bulletin is still applicable?
This FAA AC outlines how SB's should be drafted. It also says that the Design Approval Holder should make available an easy way to search the SB's

A DAH should develop a robust SB/AD tracking and management system
using the following process to ensure overlaps and conflicts between
new and existing SB actions are identified and addressed: a. Search
capabilities should include the following primary areas:
(1)
Make/model or part number of affected product(s),
(2) Major aircraft
elements (e.g., engine, strut, wing, etc.),
(3) ATA code,
(4)
Maintenance zones,
(5) Service information (e.g., service bulletin
number),
(6) Airworthiness limitation sections,
(7) Previously issued
ADs, and
(8) Planned ADs (i.e., action(s) that the FAA proposes to
mandate in an AD).

This puts the ownership on the company a bit.
The FAA makes all current bulletins available here so you can check if they still list it as current.
There is also a bit of duality between service bulletins and Air worthiness directives. SB's are issued by the company if they feel there is a safety hazard to the aircraft. Most importantly, unless they are issued alongside an AD they may not be mandatory. You can check out this blog post that goes into it a bit more.