for many proteins are specific anti-phospho-antibodies available. (and do Western blot or intracellular FACS). If there is no specific anti-phospho-antibody for your protein, you can immune precipitate your protein, run a Western blot and use for detection a general anti-phosho-tyrosine or serine antibody (check out if you expect a serine or tyrosine phosphorylation).
I'm not sure if you mean in living cells or in general. If it's not in living cells, just isolate the protein with proteinase K and phosphatase inhibitor in your lysis buffer (e.g. RIPA buffer) and then perform western blot analysis with antibodies against the phosphorylated and the unphosphorylated target. Afterwards you just have to calculate the ratio between phosphorylated and unphosphorylated protein and there you are. Sounds simple, but of course it can be a lot of work to establish the antibodies and you need a western blot imager in your lab....
You can use the techniques as mentioned by Karsten and Kristina. I would like to add some points.
First of all, if you are doing western blot analysis then try to snap freeze your protein sample after extraction. It should be done by using RIPA buffer with protease inhibitor cocktail. Then thaw the protein and do the western blot analysis.
You will get a nice signal if your protocol and protein sample and antibody is proper.
I have tried with pAKT and pERK in our lab.
Secondly, You can go for FACS analysis if you have a FACS facility in your Institute
Third, If you have proteomics facility you can do phosphoproteomics. It's an excellent technique to get phosphorylation of a downstream molecule of a signalling pathway in cells or tissues. Not only that you will get other information such as other key molecules involved in that pathway, other phospho sites, etc. This technique is little bit expensive, but you will get the whole package of data at a time.