(written from a Production point of view)
The Big Bang Theory is a CBS sitcom about four science nerds and the hot girl who lives across the hallway. The show features Brian George as Raj's father and Mark Harelik as the CalTech physics department chairman, and has a constant flow of Star Trek references.
"The Pancake Batter Anomaly"
Sheldon and Leonard play three-dimensional chess.
"The Bat Jar Conjecture"
The episode begins with Howard announcing new info on the then-upcoming Star Trek film, which leads into a discussion about alien private parts and Spock's conception. Later Sheldon is persuaded to join in the Physics Bowl by quoting Spock's dying words. Sheldon makes up TOS uniform-style shirts for his Physics Bowl team, with his teammates as redshirts. Ultimately in the final scene, to satisfy who is smarter between Leonard and Sheldon, Penny asks multiple real-world questions for which they can only give Star Trek-like answers, including Patrick Stewart or William Shatner being the Sexiest Man Alive and Tweety Bird thinking he saw a Romulan.
"The Nerdvana Annihilation"
The guys purchase a time machine prop from the movie The Time Machine and they decide to play with it. They wonder what time date to travel to and Sheldon suggests to Leonard that he should travel to the future to obtain a cloaking device and references "The Enterprise Incident".
Later, Leonard threatens to open up a Geordi La Forge action figure that's still in its original packaging.
"The Bad Fish Paradigm"
"The Codpiece Topology"
"The Panty Pinata Polarization"
"The Lizard-Spock Expansion"
Sheldon expands the game of Rock-Paper-Scissors with additional options of "Lizard" and "Spock," the latter symbolized by the Vulcan live-long-and-prosper hand gesture. It was used initially to determine whether or not to watch Star Trek: Deep Space Nine. Later, Sheldon and Raj argue whether Star Trek: The Motion Picture or Star Trek V: The Final Frontier is the worst Trek movie, and whether Star Trek II: The Wrath of Khan or Star Trek IV: The Voyage Home is the best.
"The White Asparagus Triangulation"
Sheldon compares Dr. Stephanie Barnes as the Leonard McCoy to their social group's "landing party" and gives a somewhat off version of the Vulcan salute when he notices her observing the conversation.
"The Bath Item Gift Hypothesis"
Penny gives Sheldon a used napkin autographed by Leonard Nimoy as a Christmas present. She apologizes because Nimoy wiped his mouth with it, but Sheldon is even more grateful, as now he has Nimoy's DNA.
"The Dead Hooker Juxtaposition"
Penny compares the main characters' social naivete to not knowing how to put up their shields like on Star Trek.
"The Hofstadter Isotope"
"The Vegas Renormalization"
Sheldon's frequent answer of Spock during Twenty Questions.
"The Classified Materials Turbulence"
Leonard says that Howard's zero-gravity toilet for the International Space Station will allow its crew "to boldly go where no man has gone before."
"The Electric Can Opener Fluctuation"
Penny references Star Trek as part of her comforting of Sheldon after he learns Leonard, Howard, and Raj had faked his discovery of magnetic monopoles in order to keep him happy. Sheldon wears a goatee resembling the one Spock wears in TOS: "Mirror, Mirror". A recurring joke is Sheldon's "Vulcan hearing."
"The Creepy Candy Coating Correlation"
Sheldon states he used to be a Wil Wheaton fan until Wil abruptly pulled out of a convention Sheldon had traveled ten hours by bus to attend and get his autograph. During a customizable card game tournament Wil is participating in, Sheldon constantly quotes Khan Noonien Singh from Star Trek II: The Wrath of Khan.
"The Adhesive Duck Deficiency"
Trying to get Sheldon to drive faster, Penny says, "Warp speed ahead, Mr. Spock", to which Sheldon replies that that order would never be given to the science officer, but to the navigator/helmsman instead.
"The Bozeman Reaction"
"The Excelsior Acquisition"
Sheldon reveals that Leonard Nimoy took a restraining order out against him.
"The Precious Fragmentation"
"The Pants Alternative"
"The Wheaton Recurrence"
"The Staircase Implementation"
Season 4 promotional ads
In one, Sheldon builds a TOS transporter to beam them (with Star Trek: The Next Generation-style visual effects) from their old Monday night timeslot to Thursday. In another he and Leonard again play three-dimensional chess while discussing the move.
"The Hot Troll Deviation"
"The Apology Insufficiency"
"The 21-Second Excitation"
Leonard mentions how they all waited fourteen hours to see Indiana Jones Raiders of the Lost Ark. Later, when Wil Wheaton jumps the line, Sheldon quotes Picard's "The line must be drawn HERE!" line from Star Trek: First Contact.
"The Alien Parasite Hypothesis"
"The Bus Pants Utilization"
Sheldon's passive-aggressive stance against the guys for kicking him out of their app design team includes playing the TOS theme on a theremin.
"The Love Car Displacement"
Leonard uses the Romulan Neutral Zone as a metaphor for how he and Penny will share a bed without becoming intimate. When he suggests they do get intimate in the context of how often the Neutral Zone was entered by both the Federation and Romulan Star Empire, Penny says she's raised shields over the lower half of her body.
"The Thespian Catalyst"
"The Toast Derivation"
Sheldon invited LeVar Burton over to the apartment via Twitter. At the end of the episode, he sticks his head in and gets too weirded out by the sight of three other guys singing karaoke. He then leaves and says he's so done with Twitter.
"The Prestidigitation Approximation"
Leonard says to Priya that the only Star Trek uniforms he has are the everyday and dress ones from the Original Series, and then she reveals that Raj has an Uhura dress when Leonard suggests she wear one.
"The Zarnecki Incursion"
Sheldon carries a bat'leth with him to the house of the guy who hacked his World of Warcraft account.
"The Agreement Dissection"
Sheldon cites "Let That Be Your Last Battlefield" as his inspiration for blackmailing Leonard into signing a new roommate agreement that favors him. He then shows an okudagram-styled auto-destruct-esque countdown timer on his laptop showing how much time Leonard has before Priya's parents are e-mailed about her dating Leonard. Sheldon then tells her that while she may have gone to the Faculty of Law at Cambridge University, he's an honorary graduate of Starfleet Academy. There is no "Law School". This was clearly an American writing this. It is the Faculty of Law at Cambridge University.
"The Wildebeest Implementation"
"The Roommate Transmogrification"
Raj overhears Leonard and Priya having a TOS-themed sex fantasy.
"The Boyfriend Complexity"
Howard and Raj play a game of Star Trek-themed battleships. When Howard notices that two squares are a hit but the one in the middle is a miss, Raj comments that his Starship has a hole in the middle. When asked what starship has a hole in the middle, Raj says "a Romulan battle bagel."
"The Russian Rocket Reaction"
Wil Wheaton throws a party at his house, which has a painting of him as Wesley Crusher in uniform and a USS Enterprise-D model. Raj says he talked with Alice Krige's gynecologist, referring to Alice as "The Borg Queen From First Contact." Wil gives Sheldon an autographed still-in-package Wesley action figure, but Brent Spiner promptly opens it upon seeing it. Brent then offered Sheldon a autographed still-in-package Data action figure to make up for it but Sheldon declines and declares Brent his new mortal enemy.
"The Rhinitis Revelation"
There's a TNG tricorder displayed on the bookcase behind the couch.
"The Recombination Hypothesis"
"The Vacation Solution"
Bernadette had rejected Howard's suggestion to have wedding invitations in Klingonese.
"The Rothman Disintegration"
Kripke has Sheldon explain Rock-Paper-Scissors-Lizard-Spock to him at least three times to mess with him. Sheldon later proposes a trivia contest with Star Trek: The Original Series, Star Trek: The Next Generation, Star Trek: Deep Space Nine, and Star Trek: Voyager being four of the five categories.
"The Weekend Vortex"
Amy mistakenly refers to the Star Wars video game the guys are playing as a Star Trek video game. Penny corrects her, saying the guys don't like when the two franchises are mistaken for one another, and even suggesting there is no difference between the two.
"The Transporter Malfunction"
Penny buys Sheldon and Leonard mint-in-box 1975 Star Trek transporter toys. However they refuse to play with it, as it would lose its value. That night Sheldon dreams his 1975 Mr. Spock action figure (voiced by Leonard Nimoy) declares it illogical to own a toy that he doesn't play with. After Sheldon accidentaly breaks his toy (and replaces it with Leonard's) he later has a nightmare about being chastised by Mr. Spock about his actions and being attacked by a Gorn.
"The Hawking Excitation"
"The Stag Convergence"
Sheldon tells Wil Wheaton (who said he was attending Howard's bachelor party because it was a welcome alternative to George Takei's hot tub parties) to "talk to the hand" and gives him the Vulcan salute.
"The Launch Acceleration"
Howard tells Raj "You watch Star Trek, I live it" when discussing Howard's aborted mission to the International Space Station. Leonard and Sheldon again play three-dimensional chess. Later Leonard comes home to find Sheldon and Amy playing "Star Trek doctor" with Sheldon uncomfortably lying on the couch while Amy, dressed in a Sciences division blue TOS women's uniform, pretend-scans him with a toy Medical tricorder.
"The Countdown Reflection"
Despite his initial offer to do so being soundly rejected, Sheldon performs part of Howard and Bernadette's wedding in Klingonese and says "and the Klingon High Council" when pronouncing them man and wife.
When the rocket is launched, Sheldon quote the title sequence from the original series saying "Boldly go, Howard Wolowitz".
"The Habitation Configuration"
Sheldon tries to do a segment on Trek flags on his "Fun With Flags" web show, first with Wil Wheaton (who keeps trying to do a bit on the Federation flag much to Amy's dislike) and later with LeVar Burton, telling his viewers to "set phasers to fun." To get Amy to see things his way in the matter, Sheldon gives her the complete TNG DVD box set and briefly describes "Encounter At Farpoint" to her through her closed apartment door. Wil's house number is 1701, the Enterprise 's hull number.
"The Fish Guts Displacement"
Amy suggests that Sheldon punish her by not letting her see Star Trek Into Darkness.