Silk

If you highly value the shirt...yeah the dry cleaners is the best idea. They might be able to tell you up front how successful or not they would be.
Oily stains on cotton and silk are something I just treat at home with liquid laundry detergent rubbed directly on the stain...sometimes followed by liquid stain remover (Shout). It seems to get out most stuff. I air dry the suspect items as the clothes dryer heat might really set the stain.

sometimes thats just it when you get oil on it...try taking it to a cleaners and see what they say.

Use dawn dish detergent. Works wonders

I'd call a local Dry Cleaner or take it there for laundering if it's a really important piece of clothing.

usually, with greasy items, dish washing detergent or shampoo are recommended to tackle the grease, but silk is a different beast, so I'd at least stop in some place and ask or give them a call.

hair spray - just spray it on and let it dry then launder according to directions

I would use the Get Clean liquid laundry detergent from Shaklee. Dab some on there and wash! Shaklee's laundry detergent is natural and so gentle, yet it really does the job. I've gotten stains out of clothes before that dry cleaners couldn't. The laundry detergent, Nature Bright Laundry Booster & Stain Remover (60% more effective than Oxyclean and safe), and the Basic-H2 Organic cleaning concentrate will all do wonders for any stained clothing. Check them out at www.greennutrition.info and register to win $200 of free products at www.freegreenproducts.info. Good luck!

Try plain baby powder or corn starch to soak up the oil. One time I splattered oil all over a silk dress and this did the trick. If it doesn't work, then try the cleaners.

Try dawn dishwashing liquid and let it soak in the sink.