The gravitational-wave (GW) sky may include nearby pointlike sources as well as stochastic backgrounds. We perform two directional searches for persistent GWs using data from the LIGO S5 science run: one optimized for pointlike sources and one for arbitrary extended sources. Finding no evidence to support the detection of GWs, we present 90\% confidence level (C.L.) upper-limit maps of GW strain power with typical values between 2−20×10−50   strain2  Hz−1 and 5−35×10−49   strain2  Hz−1  sr−1 for pointlike and extended sources, respectively. The latter result is the first of its kind. We also set 90\% C.L. limits on the narrow-band root-mean-square GW strain from interesting targets including Sco X-1, SN 1987A and the Galactic center as low as ≈7×10−25 in the most sensitive frequency range near 160 Hz.