I know that there's probably a higher chance of having a neutron star that has its magnetic axis inclined to the rotational axis rather than having it perfectly aligned. If they are not aligned, the neutron star will create beams of radiation that sweep through space like the beams of light from a lighthouse. But what will happen if they were to be aligned?
It is believed that old pulsars may have their rotational axes closely aligned with their magnetic field. This would happen over a timescale of $\tau\sim10^7$ years (Lyne & Manchester (1988)). There are three sets of phenomena driving the dynamics of the alignment (Casini & Montemayor (1998)):
- Short-term ($\sim50$ days) variations caused by glitches
- Intermediate increases of the alignment angle $\theta$, moving the magnetic dipole moment towards the equator
- A set of long-term ($\sim10^7$ years) dynamics leading to a decrease in $\theta$
Only in older pulsars does this last set of dynamics dominate. What this means is that the pulses from the neutron star would appear wider than pulses from neutron stars with significant misalignment. We also would not see interpulses, signals from the opposite magnetic pole, which we see in a number of young (or simply short-period) pulsars; we would just observe one pulse per rotation period.
I will add that Lyne & Manchester identified some young pulsars (including PSR 1800-21 and PSR 1823-13) that exhibited near alignment, a possible indication that the initial offset between the two axes may have a uniform distribution. Therefore, there should be examples of young pulsars with magnetic and rotation axes almost aligned. I would assume the intermediate-timescale mechanisms would then lead to misalignment before the long-term dynamics began to dominate.
NICER observations of PSR J0030+0451 in x-rays show hot spots clustered near one pole. The hot spots are presumed to be the termination of the active magnetic field lines, so there is really no magnetic "axis". The field is more complicated.