Molecules with many polar bonds are soluble in polar solvents.Also, molecules with none or few polar bonds (many non-polar bonds) are soluble in non-polar solvent. e.g Water is a polar solvent so substances with many polar bonds are soluble in it.
Molecules can be polar or non-polar; bonds are what hold molecules together, but they are not in themselves polar or non-polar. I should point out that the most famous polar molecule in the world, the water molecule, does have covalent bonds.
There are several types of covalent bonding and those are stated below along with an example. Polar molecules with polar bonds - Hydrogen Chloride Non-polar molecules with polar bonds - Sulfur Hexafluoride Non-polar molecules with non-polar bonds - Ozone