The first example looks plain wrong to me. (And sort of Germanized, what with the random abbreviation just stuck in the middle there.) The e.g. is completely superfluous and the sentence is fine, in fact much better, if you just take it out.

The second example looks good to me.

It's an abbreviation of "for example", but you only use it to introduce examples of something you've already mentioned, such as the "various techniques" in your second example.

So you could say:
This diagnostic device has many possible applications, e.g. drug tests, or pathogen and allergen diagnositics.

Makes sense, because it refers to the object of the sentence.

But not:
Possible applications of this device are, eg. drug tests ....

Doesn't make sense, there's no object to refer to.

But I'm no grammar expert, I'm just going by Sprachgefuehl.

The punctuation of e.g. varies. Note that in AE, most writers use a period after each letter(e.g.). A single period (eg.) is always wrong. Perhaps BE usage is different, however.

Because it is always used (as Jones pointed out) to provide examples of some already named category (in other words, in apposition to noun), it is set off from the name of that category with a comma. In the following example, the general category is "possible applications" so you follow that term with e.g. and then the list of examples.

If you do not name the general category, as in your first (incorrect) example "Possible applications are e.g. drug tests or pathogen and allergen diagnostics," you should change the sentence to read "Possible applications include drug tests or pathogen and allergen diagnostics."

Since e.g. is a Latin abbreviation that translates to "for example," many writers also put a comma after the abbreviation: "This diagnostic device has many possible applications, e.g., drug tests, or pathogen and allergen diagnositics." This rule follows logically because if one writes out the words "for example" in a sentence, it is always set off with commas. So it makes sense that the abbreviation would have the same punctuation.

But I think some people think that the second comma creates excess punctuation, and so do not use it. (For many years, I dispensed with the second comma, but recently I have begun inserting it.)

In summary:

1) Use the abbreviation e.g. only in apposition to a general category that you have already named.

2) Spell it with a period after both the e and the g.

3) Always use a comma immediately before e.g. (that is, between the noun to which it stands in apposition and the abbreviation itself).

4) Optionally (and recommended), use a comma imedicately after e.g.

5) if the sentence does not end with the list of examples, then another comma should be used after the last item in the list.

6) Never end your list of examples with "etc." The abbreviation e.g. has already advised your reader that you are providing several examples. Adding etc. at the end of the list is redundant.

7) Keep in mind that it is never wrong to simply spell out the words "for example." Doing so has the advantage of eliminating guesswork about the correct punctuation.

I think I disagree with Sharper's rule #6. The abbreviation "etc." means "and the rest", and in general usage (I think) can also mean "and so forth". But "and so forth" in the context of a list of examples ("e.g. a, b, c...") is not redundant, if the examples indicate a possible progression: "these cars come in all possible colors, e.g. blue, green, yellow, banana, seafoam green, etc."

@ 49RL8: By saying/writing "for example" or "i.e.", one is already indicating that the list that follows is incomplete and only a sampling of the possibilities. For that reason adding "etc." at the end of the list is redundant.

If one wanted to make a complete list of the choices, one could introduce the list with "specifically" or perhaps "i.e." and then provide the list.

I also thought of another rule:

8) Use "i.e." only in written texts. When speaking, always say the words "for example."

@ Sharper: "i.e."? Doch "e.g.", oder? In der Schule haben wir gelernt, daß "i.e." (id est) für "that is" steht.

@alain: yes, you are right. I meant to write "e.g." Earlier in my posting I had mentioned the use of "i.e." as compared to "e.g." and must have been distracted by my 2-year old daughter who was sitting on my lap watching a movie as I typed.

Thus rule #8 should read: Use "e.g." only in written texts. When speaking, always say the words "for example."

But the same rule applies for "i.e." When speaking, say the English words (that is) rather than spelling out the letters of the abbreviation. Saying "i.e.", however, happens less often than people saying "e.g." when they are speaking.