A standard three piece cocktail shaker consists of a body, strainer and cap. Occasionally, the strainer will become stuck to the cup making it difficult if not impossible to remove. There are several possible causes for this. A sticky residue on the inside of the cup or strainer, the contracting of the metal cup as it gets cold, or a mismatched strainer and cup combination can all cause this problem.
Soak the shaker in hot water. This is the simplest and usually the quickest way to get the top off of the shaker. If the shaker was gummed up with sticky residue the warm water will work to dissolve it. If the shaker was fused because the metal of the cup contracted around the strainer as it got cold then the warm water will loosen it up. Another thing to try is placing the shaker in a bowl of ice water for five minutes then dip the top in hot water. Sometimes the temperature difference between the top and bottom will set the strainer loose.
If the shaker is metal, tap the shaker against the edge of the bar where the strainer and cup meet. This can jar the top loose. If the shaker is glass, try to twist the top off gently. It can be difficult to get a grip on the shaker because it's slippery or wet. If this is the case rubber grip pads (available in the kitchen department of most stores) can help to grip the parts. If you don't have grip pads, use mouse pads instead.
Add cold water through the strainer, into the cup. Then run the top of the strainer under hot water. This temperature difference should help to loosen the two parts if the above methods failed. Care should be taken with glass shakers here because it is possible for them to crack or shatter due to thermal shock.