Finding musicality in sheet music
One of the challenges in finding musicality in sheet music is that the most prominent features of the notation do not directly communicate the structure of the music. For example, you may assume that one could recognise the phrasing of a melody by looking at the handy vertical bar lines and beamed notes, but unfortunately it's not so simple.
Bar lines only mark fixed periods of time, and beams can indicate melodic structure, but are also broken following certain rules relating to the time signature. The natural figures and phrases of a melody need to be recognised by looking for patterns in the notes, and it its often necessary to look across the bar lines to read them correctly.
Take the following melody for example, it is a welsh folk tune and I've used slurs to indicate how I intuitively perceive the grouping of this melody when I hear it.
Y Coroni

Many of these structures cross the bar line, and none of these groups would be indicated in the notation clearly without these phrasing slurs. Knowing where these groups are is essential to know how the melody should be grouped and emphasised in performance. so how do you recognise them?
Do you notice that the notes in these groups are closely spaced with regards to their intervals, and mostly fall into simple patterns like scale runs? These features are the key, and we will explore how to recognise them using annotated examples.
The basics of recognising phrases in sheet music
Phrases in music commonly consist of a series of notes that have a shorter duration than the note at the end of the phrase. It is also common that phrases will end with a rest, and both of these features can be seen here:

Not all phrases can be recognised from the rhythm alone though, as in the following example. These two phrases can be recognised from their melodic structure: the first two bars follow the same melodic contour as the third and fourth, but the pattern is shifted down by a note in the scale, and that shift differentiates it.
The two phrases individually sound coherent because notes all follow a smooth contour, and the leap between them also makes them sound separate. Melodies that take a pattern and repeat it at a different pitch are very common.

Phrases may also have the same number of bars within a single song, commonly 2, 4 or 8 bars. The above examples were taken from a folk tune called Winter Ade (following), which has a consistent structure of two bar phrases.
Winter Ade

The effect of an anacrusis (pick up)
Melodies often begin with a partial bar, called a 'pick up' or 'anacrusis', functionally these notes lead into the downbeat, and make the music sound more fluid. If the first phrase in a melody starts with an anacrusis, typically all of them will. Thus, the phrases end before the bar line. This can be seen in in 'will the circle be unbroken' below:
Will the Circle Be Unbroken?
![X:218
Z:Jack Campin, http://www.campin.me.uk/
F:Jack Campin's Nine-Note Tunebook
% last edit 03-02-2013
M:4/4
L:1/8
K:C
"Phrase 1" CD|F2 F4 "Phrase 2" AG |F2 A4 \
"Phrase 3" AG|F3 D F2F2 |DC- C4
"Phrase 4" CD|F2 F4 "Phrase 5" A>B|c2 A4 \
"Phrase 6" FG|A3 F G2G2 |F6 |]](/res/abc_converter/phrasesexample3_40c08c6604fcc91afba6871b7e352ecc9721f47a.png)
Melodic and harmonic discontinuity
As has been seen in several of the previous examples, a common marker of a phrase end is a melodic discontinuity, a difference in pitch that is larger than among the previous notes within a phrase:
Lili Marlene
![X:306
Z:Jack Campin, http://www.campin.me.uk/
F:Jack Campin's Nine-Note Tunebook
% last edit 03-02-2013
M:4/4
L:1/8
K:C
"Phrase 1" z|E>E E>F G2 E2 |F>F F>c B3 \
"Phrase 2" z|D>D D>E F2 F>G|
B>A G>F E3 \
"Phrase 3" z|A2 B>c B2 A2 |A2 G2 B3 \
A|G2 F2 A3 G|
F2 E2 G3 \
E|G3 F F2 d2 "Phrase 4" |c4 G3 \
E|G3 F F2 D2 |C4 z3|]](/res/abc_converter/phrasesexample5_40c08c6604fcc91afba6871b7e352ecc9721f47a.png)
It is often also the case that a change in phrase will also correlate with a change in harmony. Harmony in music is when multiple notes are played together, it can create different tonal effects depending on which notes are used. Most western music is based on a harmonic system called a chord progression.
Chords and chord progressions show up in the notes of a melody if you know how to look for them. Let's use part of the welsh folk tune Wyres Megan as an example.
Wyres Megan

The chords that go with this fragment are G (G,B,D), C (C, E, G), and A minor (A, C, E). I've written out the notes of these chords below, and many of the notes in the corresponding bars of the melody are the same as the notes in the chord.
![X: 1
R: waltz
M: 3/4
L: 1/4
K: Gmaj
| [G3B3d3] | [c3e3g3] | [A3c3e3] |](/res/abc_converter/phrasesexamplechords2_40c08c6604fcc91afba6871b7e352ecc9721f47a.png)
Harmony also explains why you can sometimes have a large leap within a phrase without it sounding like a break in the phrase. There is a large jump at the start of the following tune, but the two notes, E and B, are both in the chord of E minor, which is also the chord of the bar.
Beth Yw'r Haf I Mi?

The shift in phrases 3 and 4 of Winter Ade mentioned before also represent a change in harmony, the first starting on a chord of E minor, and the lower pitched repetition fitting into a chord of D minor.

Harmony is discussed in more detail in the article Harmony for ocarina players.
When the grouping varies within a song
These shifts in the alignment of the phrasing can also change within a single song / tune. For example in 'Blow the man down', where the first phrase begins on a pick up, but a long note forces the melody to align with the bars. The music then returns to the original alignment in phrase three.
This isn't uncommon for melodies to have varying or otherwise irregular phrasing. Music can sound rather regimental and boring if it always remains in strict lockstep with the grouping implied by the time signature.
Blow the Man Down

Exercises
Here are some tunes without marked phrases, try to find them using the patterns shown previously.
Branle De Bourgogne

The Carnival of Venice
![X:1
K:C
M:3/4
L:1/8
G2 | A4G2 | F4E2 | F2D4 | z2D2E2 | F4G2 | A4G2 | E6- | E2 z2 G2 |
c4G2 | F4E2 | F2 D4 | z2 D2E2 | F4G2 | A4G2 | C6- | C2 z4 |]](/res/abc_converter/phrasesexercise3_40c08c6604fcc91afba6871b7e352ecc9721f47a.png)
My Last Farewell to Stirling
![X:564
M:4/4
L:1/8
K:EDor
A2 |D3 D FE D2|F3 G A3 \
B/c/|d3 A A2 GF|E3 D =C2
zE |D3 D FE D2|F3 G A3 \
B/c/|d2A2 AG E2|F2D2 D2|]](/res/abc_converter/phrasesexercise4_40c08c6604fcc91afba6871b7e352ecc9721f47a.png)
The Blantyre Explosion

Closing notes
Structure can often be seen within the notes of a melody in the patterns of notes, such as being closely spaced and following a regular pattern. Large leaps and harmonic discontinuities cause things to be perceived as separate phrases.
Once you're aware of the structure of a melody, you can make use of this information as discussed in Musicality: how to make your ocarina playing sound good. For example you may add a longer gap to separate two phrases audably.

Look out for these patterns in music, and cross-referance performances with notation, to see how a performer is grouping the notes.
It is worth noting that it is probable that different people will perceive these things differently, and this isn't objectively wrong.