Fix those unclear antecedents! (Win valuable prizes!)

As the two SAT Questions of the Day below demonstrate, the SAT grammar section throws various unclear antecedents at unsuspecting test-takers.
You really have to ask, "What in the heck is this clause or phrase truly modifying?" or "What does this pronoun refer back to?"
After the questions below is the "Pronoun Problems" section of the Hamilton College Writing Center's Seven Deadly Sins of Bad Writing.

Widely regarded as the greatest American chess player in history, Bobby Fischer’s prominence came as a child, winning the U.S. Open at the age of thirteen and becoming a grandmaster at age fifteen.

A. Bobby Fischer’s prominence came as
B. Bobby Fischer’s prominence came when he was
C. it was Bobby Fischer coming to prominence as
D. Bobby Fischer came to prominence as
E. his prominence came to Bobby Fischer as


Although campaign consultants have long known that scare tactics can win votes, only recently have psychologists and political scientists devised studies to find out whose votes they win and why.

A. they win
B. they can win
C. this wins
D. tactics like this wins
E. such tactics win


From Hamilton College's Seven Deadly Sins of Bad Writing:
The Sixth Deadly Sin: Pronoun Problems

Pronouns are useful as substitutes for nouns, but a poorly chosen pronoun can obscure the meaning of a sentence. Common pronoun errors include:

Unclear Pronoun Reference
A pronoun must refer to a specific noun (the antecedent).
Ambiguous pronoun reference creates confusing sentences.
Writers should spend time thinking about their arguments to make sure they are not superficial. (Unclear antecedent: who or what are superficial?)

A key difference between banking crises of today and of yesterday is that they have greater global impact. (Which crises have more impact?)

If a whiff of ambiguity exists, use a noun: A key difference between banking crises of today and yesterday is that today’s crises have greater global impact.

Vague Subject Pronoun
Pronouns such as it, there, and this often make weak subjects.
Pope Gregory VII forced Emperor Henry IV to wait three days in the snow at Canossa before granting him an audience. It was a symbolic act.
To what does it refer? Forcing the Emperor to wait? The waiting? The granting of the audience? The audience? The entire sentence?
Use a pronoun as subject only when its antecedent is crystal clear.

Agreement Error
A pronoun must agree in gender and number with its antecedent.
A common error is the use of the plural pronoun they to refer to a singular noun.
In the original state constitution, they allowed polygamy.
They (plural) refers to constitution (singular).
revised:
The original state constitution allowed polygamy.

It is often better to use a plural noun and pronoun than to use a singular noun and pronoun.
Note that indefinite pronouns such as each and everyone are singular.
Each student must meet his or her advisor. (correct but awkward)
Each student must meet with their advisor. (incorrect: singular noun, plural pronoun)
Students must meet with their advisors. (correct: plural noun and pronoun)