Does a Second In Command have to be landing and instrument current like a Pilot In Command does?
FAA 

This is a compilation of what I could find relevant to the question. Feel free to come up with corrections if I missed anything (I'm an EASA FCL guy)

Co-pilot Landing Currency:


Part 61: 3 takeoff/landings in previous 12 months (61.55)    
Part 135: no requirements other than those in part 61 (above)
Part 121: 3 takeoff/landings in previous 90 days (121.439)


Co-pilot Instrument Currency:


Part 61: no requirements for SIC
Part 135: 135.245 requires that SIC's must meet part 61 PIC instrument currency requirements in 61.57(c) (see below)
Part 121: recurrent flight training every 12 months, including at least 1 ILS approach to minimums


Following is the list of relevant FARs :

§ 61.55 Second-in-command qualifications   


  (b) Except as provided in paragraph (d) of this section, no person may serve as a second in
  command of an aircraft type certificated for more than one required
  pilot flight crewmember or in operations requiring a second in command
  unless that person has within the previous 12 calendar months:
  (2)
  Except as provided in paragraph (e) of this section, performed and
  logged pilot time in the type of aircraft or in a flight simulator
  that represents the type of aircraft for which second-in-command
  privileges are requested, which includes --
  (i) Three takeoffs and
  three landings to a full stop as the sole manipulator of the flight
  controls; 
  
  (d) This section does not apply to a person who is:[...operating part
  121,125,135...] 
  
  (e) The holder of a commercial or airline transport
  pilot certificate with the appropriate category and class rating is
  not required to meet the requirements of paragraph (b)(2) of this
  section, provided the pilot: [... is not carrying paying
  passengers...]


§ 61.57 Recent flight experience: Pilot in command.  


  (a) General experience.
  
  (1) Except as provided in paragraph (e) of this section, no person may
  act as a pilot in command of an aircraft carrying passengers or of an
  aircraft certificated for more than one pilot flight crewmember unless
  that person has made at least three takeoffs and three landings within
  the preceding 90 days, [...]
  
  (c) Instrument experience.
  
  Except as provided in paragraph (e) of this section, no person may act
  as pilot in command under IFR or in weather conditions less than the
  minimums prescribed for VFR, unless within the preceding 6 calendar
  months, that person has: [...6 approaches, holdings, intercepting
  &amp;tracking course...]


§ 135.245 Second in command qualifications  


  (a) [...] no certificate holder may use any person, nor may any person
  serve, as second in command of an aircraft unless that person holds at
  least a commercial pilot certificate with appropriate category and
  class ratings and an instrument rating. For flight under IFR, that
  person must meet the recent instrument experience requirements of part
  61 of this chapter.


This is an excerpt from an FAA Legal Interpretation concerning 135.245:


  Next, unlike a PIC, an SIC's qualification to operate an aircraft
  under IFR in part 135 service is not dependent on compliance with §
  135.297.  Rather, § 135.245 sets the qualifications for pilots serving as SIC in part 135 operations.  It states that a pilot must meet the
  instrument experience requirements of part 61 to serve as SIC in
  flight under IFR.  Section 61.57(c) sets out the recent instrument
  flight experience requirements which include, among other things, six
  instrument approaches within the previous six months.  See Legal
  Interpretation to Gerald Naekel, from Donald P. Byrne, Assistant Chief
  Counsel Regulations and Enforcement Division (June 18, 1991) (noting
  that although the section title refers to pilots in command, the
  instrument recency requirements of that section apply to SICs).


§ 135.247 Pilot qualifications: Recent experience.  


  (a) No certificate holder may use any person, nor may any person
  serve, as pilot in command of an aircraft carrying passengers unless,
  within the preceding 90 days, that person has—
  
  (1) Made three takeoffs and three landings as the sole manipulator of
  the flight controls in an aircraft of the same category and class and,
  if a type rating is required, of the same type in which that person is
  to serve; or [...]


§ Part 121.433 Training required  


  (c)Recurrent training. (1) No certificate holder may use any person
  nor may any person serve as a required crewmember on an airplane
  unless, within the preceding 12 calendar months --  (i) For flight
  crewmembers, he has satisfactorily completed recurrent ground and
  flight training for that airplane and crewmember position and a flight
  check as applicable; 


§ Part 121.439 Pilot qualifications: Recent experience.  


  (a) No certificate holder may use any person nor may any person serve
  as a required pilot flight crewmember, unless within the preceding 90
  days, that person has made at least three takeoffs and landings in the
  type airplane in which that person is to serve.


EASA 

FCL is very clear about it: PIC and co-pilot both need to be landing current. If holding an IR rating then specific night currency is not required. No instrument currency is required as long as the IR is valid.

FCL.060 Recent experience  


  (b) Aeroplanes, helicopters, powered-lift, airships and sailplanes. A
  pilot shall not operate an aircraft in commercial air  transport or
  carrying passengers: 
  
  (1) as PIC or co-pilot unless he/she has carried out, in the preceding
  90 days, at least 3 take-offs, approaches and  landings in an aircraft
  of the same type or class or an FFS representing that type or class.
  The 3 take-offs and landings shall be performed in either multi-pilot
  or single-pilot operations, depending on the privileges held by the 
  pilot; and 
  
  (2) as PIC at night unless he/she: 
  
  
    (i) has carried out in the preceding 90 days at least 1 take-off,
    approach and landing at night as a pilot flying in an aircraft of the
    same type or class or an FFS representing that type or class; or 
    
    (ii) holds an IR;