We carry out two searches for periodic gravitational waves using the most sensitive few hours of data from the second LIGO science run. Both searches exploit fully coherent matched filtering and cover wide areas of parameter space, an innovation over previous analyses which requires considerable algorithm development and computational power. The first search is targeted at isolated, previously unknown neutron stars, covers the entire sky in the frequency band 160'728.8 Hz, and assumes a frequency derivative of less than 4Ã—10-10''Hz/s. The second search targets the accreting neutron star in the low-mass x-ray binary Scorpius X-1 and covers the frequency bands 464'484 Hz and 604'624 Hz as well as the two relevant binary orbit parameters. Because of the high computational cost of these searches we limit the analyses to the most sensitive 10 hours and 6 hours of data, respectively. Given the limited sensitivity and duration of the analyzed data set, we do not attempt deep follow-up studies. Rather we concentrate on demonstrating the data analysis method on a real data set and present our results as upper limits over large volumes of the parameter space. In order to achieve this, we look for coincidences in parameter space between the Livingston and Hanford 4-km interferometers. For isolated neutron stars our 95% confidence level upper limits on the gravitational wave strain amplitude range from 6.6Ã—10-23 to 1Ã—10-21 across the frequency band; for Scorpius X-1 they range from 1.7Ã—10-22 to 1.3Ã—10-21 across the two 20-Hz frequency bands. The upper limits presented in this paper are the first broadband wide parameter space upper limits on periodic gravitational waves from coherent search techniques. The methods developed here lay the foundations for upcoming hierarchical searches of more sensitive data which may detect astrophysical signals.