They are both examples of an extremely common error: using commas to join whole sentences.

Example (a) is actually made up of two whole sentences (remember that a sentence has a subject and a verb):

Her name is Sarah. (subject = Sarah; verb = is)
I met her in London. (subject = I; verb = met)

You cannot join these two sentences with a comma.

Similarly, example (b) above is also composed of two sentences:

I hate that dress. (subject = I; verb = hate)
It is a terrible color. (subject = It; verb = is)

If you feel that two or more sentences make better sense when they are linked, you can join them in other ways. Conjunctions – “joining” words like because, and or but – are useful for this:

I hate that dress because it is a terrible color.

You could also try using some other form of punctuation, like a colon (:), semi-colon (;) or dash (–), to link the sentences:

Her name is Sarah; I met her in London.

The following examples of comma usage are correct:

c) With a lot of hard work, I became an expert ballroom dancer.
d) I traveled to Lima, the capital city of Peru.

Using commas to link sentences is widespread because it is not obviously “wrong”. In fact, (a) and (b) above are typical of much casual spoken English. If you use commas in this way you will not be misunderstood, and the style is fine for very informal writing such as friendly letters or e-mails. However, for more formal purposes it is definitely incorrect and will spoil the effect of a serious or professional document.

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