Here are five of the most common: for my buddy Bhuki.
Selam u'aleikum ("peace be upon you") is a proper greeting wherever Islam is practiced; reply with "Aleikum a'selam". In Türkiye, both shortened selam and the Turkish merhaba are used as casual greetings, and when telling a friend to pass your regards to another, use "selam söyle".
Inşallah literally means "God willing", though in most contexts a more accurate contextual translation would be "hopefully".
Maşallah is a bit more difficult. Roughly "may God protect (this)", it signifies the wonderfulness of a thing. You'll see Maşallah, often along with the nazar, emblazoned on everything from delivery trucks to taxis as a protection from the "evil eye". Say it in congratulation when a friend tells you great news, when you meet a newborn or see a child who's grown a lot, or (more rarely) use it in reference to that hot chick across the dance floor.
Alhamdulillah, compared to the rest, is rare in mainstream Turkish. Like "praise God" or "hallelujah" in English, it's one of those phrases the areligious rarely employ.
Bismillahirramanirahim - "In the name of God, the most high, the most merciful." Known more briefly as the Bismilla, this phrase begins many suras. Devout young believers may tell you this phrase is one of two you must commit to heart in order to become a Muslim. The less religious use it much as Americans say "oh my God!" - but as the side of this fuel tank suggests, its cultural significance is broader.
It should be noted that otherwise widespread phrases such as Allahu akbar (God is great) are used in devotional contexts and do not figure in the popular vocabulary the way those above do.