J is not sparse. In fact, J is essentially a full matrix. It has no zeros in it at all! So why are you using sparse matrix storage? And especially on such a small, tiny matrix? I think you are using sparse storage for no good reason. In fact, sparse is costing you capability, time, and even memory. Not a lot. But it is just a bit silly here.
As far as using pinv(full(J))*F, yes, that is a way to resolve the singularity issues, but it may be a poor choice. A singular matrix is usually a sign that something is wrong. Yes, you can close your eyes, pushing forth, full speed ahead. But far too often that just results in a train wreck.
As a wild guess, the use of the variable names J & F here indicates to me that you might be trying to do some sort of Newton iteration. A singular matrix there suggests that you are getting into a bad place in that search, that something is wrong. But then, what do I know, since you have told us nothing at all, except that you seem to be very confused in why you are doing what you are doing. So maybe, just maybe, it is time to slow down. Look at the warnings, and take that curve ahead with the brakes applied. (In other words, think very carefully about what you are doing.)