In each of the following five examples, I notice that native French speakers pronounce the "s" sounds in bold as "z" instead. In what instances should you use an "z" sound? Is there a pattern?
les traces d'une ancienne civilisation inconnue
parce que eux, ils se dirigent vers la montagne
On est tous venus dans le même but.
Cesse de te débattre.
aller à la rescousse de tes amis
What you're hearing is called regressive voicing assimilation, and happens whenever two consonant which differ in voicing are pronounced consecutively. It's not unique to /s/, and in fact might be slightly less common with this consonant than with some others.
Just in case, some background: voiced consonant are pronounced with the vocal folds vibrating; while they don't while you're articulating voiceless consonants. In English, stout and doubt, as well as singer and zinger, differ by their voicing.
In French there are six pairs of consonants that differ by the voicing: /f/ and /v/, /s/ and /z/, /ʃ/ (boucher) and /ʒ/ (bouger), /p/ and /b/, /t/ and /d/, /k/ and /g/. The rest of the consonants are unspecified with regard to voice: they are generally voiced, can devoice by assimilation, but don't normally voice neighbouring consonants. (/ʀ/ is voiceless in some dialects and unspecified in most others)
When two consonants with a different voicing specification follow one another, the voicing of the second one tends to invade the first one. The first consonant assimilate in voicing to the second one. This voicing assimilation goes back toward the beginning of the word, and is thus called regressive. (1)
The assimilation can be complete or incomplete (consonant takes time to articulate and the vocal folds can start or stop vibrating halfway through) and happens more the closer both consonants are to each other: assimilation will be more consistent and more complete in "Je parle pas de toi" /pattwa/ (/t/ and /d/ only differ in voicing) than in Afghan (/f/ and /g/ differ in voicing, place and manner of articulation).
While French's orthography obscures this, this is what's happening in all of OP's examples:
les traces d'une ancienne civilisation inconnue /lɛtʀasdyn/
parce que eux, ils se dirigent vers la montagne /isdiri:ʒ/
On est tous venus dans le même but. /tusvəny/
Cesse de te débattre. /sɛsdətdebatʀ/
aller à la rescousse de tes amis /ʀɛskusdətɛzami/
Under the influence of /d/ and /v/ (voiced consonants), /s/ is pronounced close to, or exactly like, its voiced counterpart [z].
The problem for a learner of French is that most native speakers are deaf to assimilation, as illustrated by the comments to this question, or to this one.
Some learner-targetted material touches on this, however: see these three pages in French 1 2 3, for a very good overview.
On the linguistic side of things, studies are many, and can get very technical (voiced and voiceless consonants can differ in more ways than vocal folds vibrations, like duration and articulator constriction), but this article (2) has an overview of (de)voicing degree in different contexts:
In careful journalistic speech, a fricative (like /s/) preceded by a vowel, and followed by a voiceless consonant (eg: il se couche /iskuʃ/) is:
Fully voiceless 24.6% of the time
Partially voiced by the preceding vowel 64.1% of the time
Fully voiced 3.6% of the time
(for simplicity's sake, I'm ignoring the remaining 7.7%)
While /s/ followed by a voiced consonant (eg: il se douche /isduʃ/):
Fully voiceless 7.9% of the time
Partially voiced by the preceding vowel 27.6% of the time
Fully voiced 52% of the time
So even in the guarded speech of professional speakers, a “s” sound can be pronounced “z” around 50% of the time. The rate of voicing would only increase in a more casual context, with its typically higher articulation speed.
(1) There is one exception to this tendency: the cluster /ʃv/, which shows progressive assimilation word-internally, so that cheval ends up pronounced "chfal" rather than "jval". This wasn't always the case, compare Quebec French Joual (/ʃəval/ -> /ʃval/ -> /ʒval/ -> /ʒwal/), or, in the other Gallo-Romance languages, Picard keval (/gval/) or Walloon tchival (/d͡ʒval/).
(2) HALLÉ Pierre A. and ADDA-DECKER Martine, "Voice assimilation in French obstruents: Categorical or gradient?" 2011
When the s is the first letter of a word => the sound is "s" (example : syntaxe)
When the s is the final letter of a word, you have to check for the next first letter, if it's a vowel => the sound is "z" (example : les avions => "les zavions")
When the s is between two letters, then the sound is always "s", except if is it two vowels (example : rose => "roze")
About c, c is pronounced "ke" if there is a, o and u after it, or a consonant.
Examples : café, côté, cuisine, classe.
If it is e, i or y, it is pronounced "se".
Examples : cercle, citrouille, cycle.
Hope it helps.
