There can be no closer white dwarf. The coolest, oldest white dwarfs (3000K), would be rare, but are still luminous enough $6\times10^{-6} L_{\odot}$ to have been easily detected at distances closer than Sirius. At the distance of Sirius, such an object would have a visual magnitude of around 12-13 and would be brighter at near infrared wavelengths where all sky surveys such as 2MASS would definitely have spotted it from its parallax.
Neutron stars and black holes could be almost undetectable but are expected to be roughly 10 and 100 times rarer respectively. Calculated as follows:
Let us assume that $N$ stars have ever been born in the Milky Way galaxy, and given them masses between 0.1 and 100$M_{\odot}$. Next, assume that stars have been born with a mass distribution that approximates to the Salpeter mass function - $n(m) \propto m^{-2.3}$. Then assume that all stars with mass $m>25M_{\odot}$ end their lives as black holes, all stars with $8<m/M_{\odot}<25$ end their lives as neutron stars and about half the stars with $0.9<m/M_{\odot}<8$ end their lives as white dwarfs (the other half are still alive as main sequence stars, as are all stars born with lower masses).
So, if $n(m) = Am^{-2.3}$, then
$$N = \int^{100}_{0.1} A m^{-2.3}\ dm$$
and thus $A=0.065N$.
The number of black holes created will be
$$N_{BH} = \int^{100}_{25} Am^{-2.3}\ dm = 6.4\times10^{-4} N$$
i.e 0.064% of stars in the Galaxy become black holes. NB: The finite lifetime of the galaxy is irrelevant here because it is much longer than the lifetime of black hole progenitors.
In a similar way, the number of neutron stars
$$N_{NS} = \int^{25}_{8} Am^{-2.3}\ dm = 2.6\times10^{-3}N$$
and the number of white dwarfs
$$N_{WD} = 0.5\times \int^{8}_{0.9} Am^{-2.3}\ dm = 0.027 N$$
Now we use these results as scaling factors to apply to the local stellar population. There are about 1000 "normal" stars in a sphere of 15 pc radius, thus a density of 0.07 pc$^{-3}$. Thus one uses the results above to calculate the density of compact remnants and then take $(3/4\pi n)^{1/3}$ as an estimate of the average distance to one of them. This gives an expectation value of 18 pc to the nearest black hole, 11 pc to the nearest neutron star and 5 pc to the nearest white dwarf.
Thus the distance to the nearest white dwarf is roughly as expected. For reasons discussed in my answer to this related question the distance calcuated to the nearest black hole and neutron star remnants is likely to be an underestimate because many escape from the Galaxy or have very high velocity dispersions and much larger Galactic scale heights than normal stars. So whilst it is possible that an unseen one exists closer than Sirius, it is highly unlikely.
How could such an object be detected? An old, cold neutron star or black hole could be completely undetectable at all wavelengths of electromagnetic radiation - though it could be fruitful to examine carefully any candidate detections [see below] for signs of X-ray emission due to accretion from the interstellar medium). But your question has I think the correct suggestion. The objects would likely have a substantial proper motion and so there is a decent chance that you would see a "moving" gravitational lensing signature. This would still be very small unless the object just happened to pass directly in front of a background star - but such a microlensing event would be transient and may not be observed. More likely is that Gaia would pick up the subtle shifts in the positions of background stars changing over the 5 years of its mission. As per your other question: Will Gaia detect inactive neutron stars?
