Context. The discovered Uranus Trojan (UT hereafter) 2011 QF 99 , as well as several candidates, has been reported to be on unstable orbits. This implies that the stability region around the triangular Lagrange points L 4 and L 5 of Uranus should be very limited. Aims. In this paper, we aim to locate the stability region for UTs and find out the dynamical mechanisms responsible for the structures in the phase space. The null detection of primordial UTs also needs to be explained. Methods. Using the spectral number as the stability indicator, we construct the dynamical maps on the (a 0 , i 0 ) plane. The proper frequencies of UTs are determined precisely with a frequency analysis method so that we can depict the resonance web via a semianalytical method. We simulate the radial migration by introducing an artificial force acting on planets to mimic the capture of UTs. Results. Two main stability regions are found, one each for the low-inclination (0 • -14 • ) and high-inclination regime (32 • -59 • ). There is also an instability strip in each of them, at 9 • and 51 • respectively. They are supposed to be related with g − 2g 5 + g 7 = 0 and ν 8 secular resonances. All stability regions are in the tadpole regime and no stable horseshoe orbits exist for UTs. The lack of moderate-inclined UTs are caused by the ν 5 and ν 7 secular resonances, which could excite the eccentricity of orbits. The fine structures in the dynamical maps are shaped by high-degree secular resonances and secondary resonances. Surprisingly, the libration center of UTs changes with the initial inclination, and we prove it is related to the quasi 1:2 mean motion resonance (MMR) between Uranus and Neptune. However, this quasi resonance has ignorable influence on the long-term stability of UTs in the current planetary configuration. About 36.3% and 0.4% of the pre-formed orbits survive the fast and slow migrations (with migrating time scales of 1 and 10 Myr) respectively, most of which are in high inclination. Since the low-inclined UTs are more likely to survive the age of the solar system, they make up 77% of all such long-life orbits by the end of the migration, making a total fraction up to 4.06 × 10 −3 and 9.07 × 10 −5 of the original population for the fast and slow migrations, respectively. The chaotic capture, just like the depletion, results from the secondary resonances when Uranus and Neptune cross their mutual MMRs. However, the captured orbits are too hot to survive till today. Conclusions. About 3.81% UTs are able to survive the age of the solar system, among which 95.5% are on low-inclined orbits with i 0 < 7.5 • . However, the depletion of the planetary migration seems to prevent a large fraction of such orbits, especially for the slow migration model. Based on the widely-adopted migration models, a swarm of UTs at the beginning of the smooth outward migration is expected and a fast migration is in favour if any primordial UTs are detected.