This an ancient problem, and the solutions falls into various categories like:
- Stop the origin of the sound – Talk to the neighbour...
- Block out the sound – Use ear plugs, noise canceling headphones, reinforce walls, building a (more or less) soundproof cubicle around your bed
- Move further away from the sound – If possible, change the room you sleep in, to a room further away from the sound source
- Introduce intermediate sound – See below...
After being diagnosed with a mild variant of sleep apnea I received a CPAP (Continuous Positive Airway Pressure) apparatus. This had the side effect of creating a little local noise which helped me eliminate other noise sources.
In other words, if neither of the other options above is able to help you, and none of the variants in this thread, it could be an option to introduce an intermediate sound source in your bedroom. Kind of like white-noise, but really any sound raising the ambient sound level will make other sound vanish to some extent.
It'll require some getting used to, but when done you've trained yourself to sleeping in a noisier environment, and you'll benefit from it in other surroundings.
A last note related to ear plugs (or half ear plugs if they annoy you when sleeping), there are other alarms not relying on sound to awaken you. For example they could use vibration or light. But note, that most people will awaken from the local alarm even though accustomed to sleeping in a noisy environment.