Next code prints (at the Python Console of QGIS) indices (from point and line layers) and minimum distances for each feature involved in each layer.
registry = QgsMapLayerRegistry.instance()
points = registry.mapLayersByName('Random points')
lines = registry.mapLayersByName('lines')
feat_lines = [ feat for feat in lines.getFeatures() ]
feat_points = [ feat for feat in points.getFeatures() ]
for i, feat_point in enumerate(feat_points):
distances = 
for j, feat_line in enumerate(feat_lines):
print i, distances.index(min(distances)), min(distances)
I tried out my code with shapefiles of next image:
All values were corroborated and they were as expected. For convenience, it is preferable to add these values to the attributes table of points layer instead of the line layer.