  Nova explosions are caused by global thermonuclear runaways triggered in the
surface layers of accreting white dwarfs. It has been predicted that localised
thermonuclear bursts on white dwarfs can also take place, similar to Type I
X-ray bursts observed in accreting neutron stars. Unexplained rapid bursts from
the binary system TV Columbae, in which mass is accreted onto a
moderately-strong magnetised white dwarf from a low-mass companion, have been
observed on several occasions in the past $\approx40$ years. During these
bursts the optical/UV luminosity increases by a factor of $>3$ in less than an
hour and fades over $\approx10$ hours. Fast outflows have been observed in UV
spectral lines, with velocities $>3500$ km s$^{-1}$, comparable to the escape
velocity from the white dwarf surface. Here we report on optical bursts
observed in TV Columbae as well as in two additional accreting systems, EI
Ursae Majoris and ASASSN-19bh. The bursts have a total energy $\approx~10^{-6}$
those of classical nova explosions ("micronovae"), and bear a strong
resemblance to Type I X-ray bursts. We exclude accretion or stellar magnetic
reconnection events as their origin and suggest thermonuclear runaway events in
magnetically-confined accretion columns as a viable explanation.
