You should usually avoid starting sentences with However for three reasons.
First, writers who start one sentence with However will often start many sentences with However. This repetition is ineffective because it serves no purpose (repetition should be used intentionally for emphasis) and therefore creates a dull, mindless pattern. The repetition of However in mid-sentence can also be a problem but is a less serious problem because the repetition is not so obvious.
Second, however should be positioned to emphasize the words that precede it. Consider the following examples.
1. The public believed the report. However, scientists remained uncertain.
2. The public believed the report. Scientists, however, remained uncertain.
Note that the location of however in sentence 2 strengthens the difference between the public and scientists. So, However can often provide greater emphasis when located after subject.
Third, However is a slow way to start a sentence. Many excellent writers prefer to start with But because it is more direct. Unfortunately, some ignorant reviewers and editors think that starting sentences with But is incorrect. So, if you use But at the start, be ready to fight. Note that starting many sentences with But or any other word will create the same problem of mindless repetition mentioned earlier.
Although you should avoid starting sentences with However, doing so does not break any rule of grammar, and some sentences may be better if the However is located at the start. But do not start several nearby sentences with However and do not miss the opportunity to provide useful emphasis by locating However later in the sentence. As always, you will need to listen to your sentences and paragraphs to determine how well they are working.
The following version of sentence 1 is common but incorrect:
3. The public believed the report, however, scientists remained uncertain.
Because of incorrect punctuation, the reader doesn't know whether the however goes with the first or second clause in sentence 3. And therefore sentence 3 would be considered a run-on. To correct sentence 3, place a period or semicolon in front of however:
4. The public believed the report; however, scientists remained uncertain.
But now the second clause in sentence 4 starts with however. So let's try some alternatives.
5. The public believed the report, but scientists remained uncertain.
6. Although the public believed the report, scientists remained uncertain.
Note that the but in sentence 5 can be preceded by a comma, because unlike however, which is a conjunctive adverb, but is a coordinating conjunction. Coordinating conjunctions (and, but, or, nor, so, yet) link parallel elements into one sentence. Conjunctive adverbs cannot be used to link parallel elements. Here are some other conjunctive adverbs: furthermore, thus, therefore, consequently.