Dear Rachel Lee:

You wrote:
>> Is there any difference in meaning in the following sentences and are
>> there any alternatives which convey the similar meaning?
>> 1. He washed his hands for fear that he should be contaminated.
>> 2. He washed his hands for fear that he would be contaminated.
>> 3. He washed his hands for fear that he might be contaminated.
>>
All three are pretty similar. “Should” is a little stronger than “would” in most cases; similarly, “might” tends to be weaker than “would” in terms of likelihood of the event happening, but the distinction is slight in most cases.

There are certainly other ways of saying this, e.g., “He washed his
hands for fear of being contaminated.”

I hope this helps.