The NBA’s Chicago Bulls had two games postponed with 10 of the team’s players in the league’s health and safety protocols. The English Premier League reintroduced what it called “emergency measures” as giants Manchester United and Tottenham Hotspur had games postponed and their training facilities shuttered because of outbreaks. The Calgary Flames became the third NHL team to have their season paused because of an outbreak.

The NFL, meanwhile, had 37 players test positive on Monday—the league’s highest single-day total ever, a reflection of weekly testing and rising cases in the league. A staff member of the Washington Football Team was also identified as having the new Omicron variant. Then, after playing on Monday night in Arizona, the Los Angeles Rams decided to work virtually on Tuesday and Wednesday because of a rapidly growing outbreak inside the club on a day when there were at least two dozen more positives across the sport.

There is much less alarm for the leagues as 2022 approaches than there was in early 2020. Players and coaches are vaccinated. Stadiums are packed with fans. The games will go on. Yet testing still turns up cases every day, sidelining players or entire teams.

The NFL might have the most at stake. Weeks away from its playoffs, an inconveniently timed cluster of positives could tilt the balance of its season—and might even decide which teams play in the Super Bowl.

Sports leagues became billion-dollar laboratories during the pandemic, and they poured their vast resources into figuring out how to shoot, throw and kick balls even as the virus raged around them. The NBA took the most extreme step when it moved into a bubble. The NFL survived its 2020 season without missing any games by pouring tens of millions of dollars into testing and contact tracing. The leagues wrote and rewrote cumbersome policies around masking, social distancing and close contacts to prevent the virus’s spread.

Vaccines, which studies have shown to be safe and effective at preventing severe illness from the virus, changed the paradigm in 2021. More than 94% of NFL players and every high level staff member has been vaccinated, according to the league. That number is 97% in the NBA, where 60% of eligible players are boosted. The leagues adjusted their protocols accordingly—reducing the testing frequency and strict rules for those who have gotten the shot.

Those vaccines have been critical for establishing the relative ease the leagues have experienced in recent months. NFL officials, for instance, have said they have not seen the same unchecked spread among players and staff that they did a year ago. But between the small number of unvaccinated players and so-called breakthrough cases, there can be enough to stir trouble.

Across the Atlantic, Europe appears to be a preview of things to come, just as it was during the first wave of virus in 2020.

German soccer earlier this month returned to playing matches in front of limited crowds. Some are back to empty stadiums. The English Premier League isn’t there yet, but clubs have seen a spike in cases among players and coaches as the Omicron variant spreads throughout the UK. The league last week reported 42 new cases—the highest weekly positive count since the beginning of the pandemic. It was unsurprising when match postponements and stricter protocols followed close behind.

Tottenham Hotspur’s clash with Brighton and Hove Albion was the first to fall off the schedule last Sunday, before Manchester United’s Tuesday game against Brentford had to be scrapped due to an outbreak within the United squad. Spurs and United also shut down the teams’ practice facilities.

English soccer’s immunization numbers suggest that the disruptions won’t end there. The Premier League, which encourages vaccination but hasn’t made it mandatory, most recently said that 68% of its players were fully jabbed as of mid-October.

The NBA postponed games this week for the first time this season in an ominous callback to the league’s Covid surge last winter. The league had been able to avoid disruptions until now, even as dozens tested positive with breakthrough infections, but that’s quickly changing.

More than a third of the league’s teams have players sidelined by the NBA’s health and safety protocols. No team has been slammed harder than the Chicago Bulls, who were down 10 players when the league postponed their games scheduled for Tuesday and Thursday, a response to a rare outbreak in a fully vaccinated, mostly boosted locker room. The NBA tightened its testing schedule after Thanksgiving, which resulted in an uptick of positives and identified asymptomatic cases.

The league’s vaccinated players don’t have to be quarantined after exposure and only have to be isolated if they test positive for Covid. One case was theoretically enough to wipe out an entire team for a few days last season. This season it takes something like the Bulls’ cluster.

In the NHL,  the Ottawa Senators, New York Islanders and Calgary Flames have had outbreaks among fully vaccinated players that spread so widely that the league announced the teams would halt play. Other teams have teetered on the edge as significant numbers of players entered Covid protocols.

Epidemiologists say there’s little mystery why the virus can proliferate when unmasked people are regularly in each other’s faces in poorly ventilated ice rinks, even after vaccination. The Covid testing protocols devised by the league in September for the era of full vaccination also likely allowed individual cases to ripple through teams. Those rules will be up for review at a meeting Wednesday between the league and players’ association.

The NFL has structural advantages over other leagues in its pandemic battle. Team rosters are huge, so losing a handful of players doesn’t prompt postponements—even if it can have an outsize impact on game outcomes, such as when the Green Bay Packers lost Aaron Rodgers for a game. Its teams also play just once a week, yielding more time and flexibility to manage virus snafus.

The rising case numbers in the NFL indicate how this worrisome trend is arriving at the worst time. The end of the regular season is close. A Rodgers-like twist in the playoffs could affect the outcome of the Super Bowl.

The NFL called an audible Monday. While a new memo to teams said that unvaccinated players have tested positive at a much greater rate than those who have gotten the shot, it echoed the CDC’s guidance calling for those eligible to receive booster shots. The memo stopped short of requiring players to get boosted, but it said eligible “Tier 1 and Tier 2 staff”—figures such as coaches and others who interact with players—have to get boosters within the next two weeks.

—Louise Radnofsky and Joshua Robinson contributed to this article.

Write to Andrew Beaton at andrew.beaton@wsj.com and Ben Cohen at ben.cohen@wsj.com