Birds have integrated active control systems that rely on several sensors to adjust the control surfaces to stabilize the flight. Since it is very difficult (perhaps impossible) to study the aerodynamic properties of the bird independently from its control system it is hard to make any claims on the static or dynamic stability of a bird.
What can be said is that from a control system point of view, most birds are observable, controllable and stabilizable in flight. This does not apply to the class of flightless birds.
If you are interested in the similarities in aerodynamics between insects, birds and aircraft I recommend getting a copy of "The Simple Science of Flight; from insects to jumbo jets" by Henk Tennekes, an Emeritus Professor of Aerospace Engineering at Pennsylvania State University.
After sitting in the garden on a sunny afternoon, observing various bird species and tinkering a bit more on the subject I actually think that many birds are unstable. Especially the smaller species that fly in the garden through bushes and trees, but also those that hunt for flying insects, are capable of extreme rapid changes in direction of flight; they are very agile.
Stability and extreme manoeuvrability are difficult to combine. In fact modern jet fights are designed to be unstable to improve their manoeuvrability. They then need a computerized control system to stabilize them.
An online search showed that some research has been done on the field of flight stability of birds Petter Krus of the Linköping University in Sweden has published an article on Natural methods for Flight Stability in Birds.
There is a stark contrast between the behaviour of flying man-made
artifacts and the effortless elegance of bird flight. In this paper,
mechanisms are proposed that seems to be at least related to the way
birds fly when they are gliding (the no propulsion case). It is
recognized that biological structures and actuator systems (muscles)
are characterized by a great deal of compliance that produce a
completely different dynamic behaviour of flight. In this paper it is
demonstrated how this can be applied on a geometrically unstable
bird-like configuration. The result is that geometrically unstable
configurations can be stabilised without the need of rate gyros and/or
accelerometers. Although the bird might have a fast acting control
system for flight control, it is unlikely that it is fast enough for
active stabilisation of the short period oscillation, and this is
unnecessary with the model described here, which only requires
stabilisation of the long period oscillation.
In his introduction he also briefly refers to the work of Tennekes, which does not include much about the stability of birds. Also in the introduction he observes that birds have a cambered wing and must have an aft centre of gravity. This combination makes the bird unstable (if it were a rigid body).