What me know what I'm talking about, just commiting a bit of original research and shouting history!
Anyway I did a bit of digging in  google which turned up
Gregory St. Vincent (Grégoire de Saint-Vincent), with -
The Greeks described a spiral using an angle and a radius vector, but it was St. Vincent and Cavalieri who simultaneously and independently introduced them as a separate coordinate system. In an article The Origin of polar coordinates, J. J. Coolidge refers to the priority dispute between Cavalieri and St. Vincent over their discovery. St. Vincent wrote about this new coordinate system in a letter to Grienberger in 1625 and published the process in 1647. On the other hand Cavalieri's publication appeared in 1635 and the corrected version in 1653.
and Milestones in the History of Thematic Cartography
Development of the use of polar coordinates for the representation of functions. Newton's Method of Fluxions was written about 1671, but not published until 1736. Jacob Bernoulli published a derivation of the idea in 1691. [238,p. 324] attributes the development of polar coordinates to Fontana, with no date.- Isaac Newton  (1643-1727), England, and Gregorio Fontana  (1735-1803) and Jacob Bernoulli  (1654-1705) [238,p. 324]. 1671 is probably too early; 17361755 would probably be more appropriate. There are earlier references to Hipparchus (190-120BC) regarding the use of polar coordinates in establishing stellar positions, and Abu Arrayhan Muhammad ibn Ahmad al-Biruni (1021) regarding the use of three rectangular coordinates to establish a point in space.
1843 Use of polar coordinates in a graph(frequency of wind directions)- Léon Lalanne (1811-1892), France [148].
La Habra High School Math History Timeline
Jacob Bernoulli invents polar coordinates, a method of describing the location of points in space using angles and distances.
Sherlock Holmes in Babylon and Other Tales of Mathematical History has a section
Newton as an originator of polar coordinates
Jacob Amsler http://www-history.mcs.st-and.ac.uk/~history/Printonly/Amsler.html
[Amsler] invented the polar planimeter, a device for measuring areas enclosed by plane curves. It was based on polar coordinates whereas earlier instruments were based on cartesian coordinates. In 1856 Amsler published a paper Über das Planimeter in which he gave details of his idea. As Mahoney writes in [1], Amsler's planimeter:- ... adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series: it proved especially useful to shipbuilders and railway engineers. 
So it seems there is considerable different accounts of their introduction, and quite a few notable applications.