Kate Pulaski joined the Enterprise in Season 2 of TNG after Doctor Crusher left to run Starfleet Medical. She only stayed a year, and while it was nice to have Crusher back afterward, it was great to have Pulaski around. Irreverent, passionate and a doctor and surgeon of prodigious skill, Kate was a valuable addition to the crew.
A lot of fandom (although not this lovely corner of it) seems to be pretty down on Pulaski. From my investigations, there are a few main reasons for this. And this is my strategy for knocking some sense into those people.
For the people who claim she has an awful bedside manner, I'd recommend watching The Child, Contagion, Pen Pals and Up the Long Ladder. For the people who say she's just plain mean to Data, take a look at The Child, Unnatural Selection, Pen Pals and Peak Performance.
And for the people who complain because she's too old and not Doctor Crusher, they are beyond help.
The Child - Kate's very first job as CMO is to diagnose Deanna's unexpected alien pregnancy, and when she has, instead of rushing to inform the captain, she takes Deanna for a drink (presumably synthehol, at least for Deanna) in Ten-Forward to talk it over. That's the sort of old-fashioned doctoring I can get behind.
She also delivers the baby and sciences in an incisive and clever fashion.
And although she teases Data about his name, she makes an effort to get it right and takes the time to explain idioms to him.
Where Silence Has Lease - I blame this episode for Pulaski's bad reputation. It's only her second episode and the only thing she gets to do in it is to insult and demean Data. However, in her defense, nobody is in character in this one. This is the episode where Captain Picard decides, after exploring one or two alternatives in a lacklustre fashion, that the best course of action when provoked is to sit around and wait for death. Nobody should be judging anyone based on this episode. The only reason I even included it is to warn people away.
Elementary, Dear Data - Although she doubts Data's capacity to solve complex mysteries by himself, it's more of a friendly challenge than a scornful dismissal, and she's willing enough to go along with the attempt and doesn't begrudge him his victory.
Plus when she's kidnapped by an out-of-control hologram she just chills and eats crumpets while looking awesome in period costume.
Loud as a Whisper - Turns out Doctor Pulaski is a pioneering eye surgeon who has done operations that people thought were impossible until she did them. This is mentioned fairly casually. No big deal, apparently.
Unnatural Selection - This episode begins with an eminent physician outright fangirling Pulaski. Which is fun. Apparently a paper she wrote about viral propagation is still the standard on the subject after several years. Pulaski has academic cred as well as being an ace surgeon.
In this episode she gets a medical mystery to solve, and she pursues it with tenacity and passion, arguing her case with the captain and even risking her life to save other people.
She and Data bond when he goes with her on a dangerous mission, and ooh, also, she hugs Picard and it's kind of adorable.
The Dauphin - Not only does she successfully defend her patient from a creature bent on killing him, she realises exactly what creature it is just from having seen it once in a zoological catalogue.
Contagion - The whole ship is in crisis! Nothing's working! The medics can't get the sickbay equipment to work! No worries, Pulaski can show you how to make a splint, as well as running Sickbay in a shipwide emergency while barely breaking a sweat.
Time Squared - Riker hosts a breakfast party and Kate brings ale. Which is weird but kind of awesome. Also she and Deanna have a disagreement about the captain and yell at each other a bit, but it's ok because they both have his best interests at heart.
The Icarus Factor - This is a big episode for Kate. She's reunited after many years with an old flame, Kyle Riker, and tries to fix his relationship with his son, although when it gets too exasperating watching their posturing, she and Deanna have to sneak off to an empty conference room to eyeroll at one another about their menfolk.
This is also the episode where we discover she's been married (and it's implied divorced, but this is the 24th century, who knows?) three times, and also that she regularly prescribes her patients chicken soup just to cheer them up a bit.
Plus, she goes to Worf's ascension day party, even though it makes her uncomfortable, just because she cares.
Pen Pals - She worries about Wesley, acting as the voice of reason and questioning the amount of responsibility he should have at such a young age. Later, she encourages him to believe in himself.
She pulls no punches when discussing her feelings about the Prime Directive - when it comes to it, she'd rather save lives than follow the rules. She urges the captain to take Data's feelings into account, and reassures Data that he's doing the right thing.
Plus, she basically invents an incredibly delicate brain surgery to remove specific memories in about five minutes flat.
Samaritan Snare - When the captain refuses to have essential surgery on the Enterprise because he's afraid of looking weak in front of the crew, Pulaski teases him about it but lets him go off to some starbase to have it anyway.
When the surgery goes wrong, she's there in time to fix it and save his life, and doesn't even mind when he acts all childish and ungrateful about it.
Up the Long Ladder - When Worf falls prey to an embarrassing illness, Kate saves him from humiliation in front of the rest of the crew, and to thank her he invites her to join him in the Klingon tea ceremony. Klingon tea is poisonous to humans, but she just takes an antidote and goes right ahead.
Later on, she comes up with a bright idea to save two civilisations, and recommends polyandry as a sensible way to organise a society.
Manhunt - Kate and Deanna joke good-naturedly about Lwaxana's pursuit of Captain Picard.
Peak Performance - In an attempt to teach Data about competitiveness, Kate sets him up for a contest which he ends up losing. Repentant, she apologises for manipulating him into playing and for hurting his feelings.
When he suffers a crisis of confidence, she and Deanna together persuade the captain that he needs help, and they both share in his victory at the end of the episode.
Shades of Gray - When Riker is infected by an unidentified microbe on an away mission, Kate beams down even though she's afraid of the transporter and Chief O'Brien is trying to mess with her head.
With Deanna's help, she successfully works out how to save Riker from death.