Stephen Murphy, a city councillor since 1997, and president the last two years, knows something about timing. He once compared the contest for the presidency to a Bugs Bunny cartoon.

“The first person that sticks their head out of their rabbit hole gets blasted,” he remarked to the Boston Phoenix in 2001.

The same goes this coming year for a possible race for the mayor’s office if its current five-term occupant is still ambling about, if with some difficulty. Thomas Menino hasn’t said if he’s running for another four years at City Hall; he’s focused instead on returning to full strength after lengthy end-of-the-year stays at Brigham and Women’s and Spaulding Rehabilitation Hospital.

Potential contenders – those who are interested in running only if Menino doesn’t – are rumored to be making phone calls about an open race. But publicly, they are keeping their heads below the political parapet.

Even if Menino runs again, the year promises excitement: In addition to the special election for US Sen. John Kerry’s seat, the 13 City Council seats are on the docket as are policy issues ranging from transportation financing to the budget that will dominate Beacon Hill as Gov. Deval Patrick enters his last two years on the job and state Sen. Therese Murray, a Dorchester native and Plymouth Democrat, comes closer to her term limit – 2015 – as Senate president.

Back at City Hall, the following is a glance at a few of the dynamics and issues that will likely crop up over the next 12 months.

School assignment: In the immediate future, plenty of eyes will be on how the mayor handles the radical revamping of the city’s unpopular school assignment system. The administration acknowledges that the current system, which divides the city into three zones and, in many cases, buses children from one neighborhood to another, must be improved. District 7 Councillor Tito Jackson told WGBH 89.7 last week that the overhaul will be one of the “defining moments” of the year and for the future of the city.

An advisory group of parents, academics, and nonprofit officials will return next week from a holiday break to continue its deliberations at a meeting on Jan. 7 at Suffolk University. The 27-member panel has been weighing several proposals, and leaning on people like a MIT doctoral student and his algorithms to help them move forward. They are charged with providing a proposal to Superintendent Carol Johnson that she can take to the School Committee.

Opposition to any changes has been poorly organized. A coalition of activists, including members of a bus union that will be financially affected if the city is able to cut busing costs, staged a protest in front of a School Committee meeting last month. About 20 people showed up on a chilly night, with a few being forced to pick up placards after a gust of wind blew them across Court Street. The protestors walked in a circle and argued that changes will somehow cause a “re-segregation” of a school system that is currently 13 percent white.

Casinos: The city continues its efforts to negotiate a deal with Suffolk Downs, which is seeking a license from the state’s Gaming Commission and a mitigation agreement with City Hall to build an East Boston resort casino. The neighborhood will vote on the proposal whenever it’s ready, unless a push materializes to take the vote citywide. Menino has maintained he wants balloting kept inside the neighborhood’s borders.

He has tapped an advisory committee to help with –- but not actually take part in -– negotiations that would result in a lucrative deal for the city. At a meeting last May, Brian Leary, the head of the committee, which met regularly over the past year, described himself as “agnostic” on gambling, according to the session’s minutes. But the McCarter & English attorney said the committee’s job is to make sure the “host community agreement” between the city and Suffolk Downs mitigates the casino’s effects on the neighborhood.

Separately, casino magnate Steve Wynn is looking at building a casino in Everett, further complicating a complex issue since only one license is available in the region that includes Boston and Everett.

Elections: Two possible tracks are set for 2013: One has Menino declining to run for reelection, likely prompting several city councillors to give up their seats and make a run for the mayor’s office, setting up a domino effect that would remake the City Council, far more so than when Michael Flaherty and Sam Yoon mounted campaigns against Menino in 2009. They were replaced by Ayanna Pressley and Felix Arroyo, two youthful and ambitious pols.

The other track has Menino running again, likely drawing but a few long shots into his orbit. That would portend a status quo year, municipally speaking, unless Michelle Wu, a South End resident who worked on Elizabeth Warren’s campaign, manages to get traction in her City Council at-large race.

And could Suzanne Lee mount another run for the District 2 seat now held by Bill Linehan? A former Boston Public Schools principal, Lee won the preliminary faceoff in 2011, but fell short in the final election. Linehan, a South Boston resident who has represented the district since 2007, won the close race and the ensuing recount.

In possible signal of her 2013 intentions, Lee last month deposited nearly $9,500 in donations, including a $100 check from Joyce Ferriabough, a longtime political consultant who has worked on Pressley’s campaigns.

The budget: Gov. Patrick releases his proposal for the state’s fiscal 2014 budget this month, and that will in turn inform the House and Senate’s versions. City Hall will be closely parsing the numbers and what they mean for the local aid from Beacon Hill that makes up a significant chunk of the city’s budget.

The state remains mired in an economic slowdown. At a recent State House hearing on next year’s fiscal picture, the Massachusetts Taxpayers Foundation predicted slow growth in the offing.

“While fiscal 2014 shows an improving revenue picture, we are still a long way from experiencing the level of revenue growth of prior economic recoveries,” said Michael Widmer, the business-backed group’s president.

Material from State House News Service was used in this report.