The gravitational-wave (GW) sky may include nearby pointlike sources as well as astrophysical and cosmological stochastic backgrounds. Since the relative strength and angular distribution of the many possible sources of GWs are not well constrained, searches for GW signals must be performed in a model-independent way. To that end 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. The latter result is the first of its kind. Finding no evidence to support the detection of GWs, we present 90% confidence level (CL) upper-limit maps of GW strain power with typical values between 2-20x10^-50 strain^2 Hz^-1 and 5-35x10^-49 strain^2 Hz^-1 sr^-1 for pointlike and extended sources respectively. The limits on pointlike sources constitute a factor of 30 improvement over the previous best limits. We also set 90% CL limits on the narrow-band root-mean-square GW strain from interesting targets including Sco X-1, SN1987A and the Galactic Center as low as ~7x10^-25 in the most sensitive frequency range near 160 Hz. These limits are the most constraining to date and constitute a factor of 5 improvement over the previous best limits.